Anti-EGFRvIII synNotch Receptor Induced Anti-EphA2/IL-13Ralpha2 CAR (E-SYNC) T Cells

Inclusion Criteria: * Inclusion Criteria for Cohort 1: 1. Age \>= 18 years. 2. Karnofsky performance status (KPS) score of \>=70. 3. All participants must have adequate organ function defined as: 1. Peripheral absolute neutrophil count \>=1000/mm\^3. 2. Platelet count \>=100,000/mm\^3 (transfusion independent, defined as not receiving platelet transfusions for at least 7 days prior to enrollment). 3. Absolute lymphocyte count (ALC) \>= 300/μL and/or Cluster of differentiation 3 (CD3) count of \>=150/μL. 4. Creatinine clearance or radioisotope glomerular filtration rate \>= 50 mL/min/1.73m\^2. 5. Total Bilirubin \<= 1.5 x ULN except for Gilbert's syndrome and 6. Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<= 3x upper limit of normal (ULN). 7. Left ventricular ejection fraction (LVEF) \>= 50% by echocardiogram or multi-gated acquisition scanning (MUGA). 8. Adequate pulmonary function, defined as no evidence of dyspnea at rest and pulse oximetry \> 92% while breathing room air. 4. Pathological criteria: EGFRvIII+ GBM from most recent surgery, confirmed by a Clinical Laboratory Improvement Amendments (CLIA)-certified lab using a next-generation sequencing panel. 5. MGMT promoter must be unmethylated or with a methylation index \< 3. 6. Must have completed at least standard of care (SOC) external beam radiotherapy (EBRT) as initial therapy. 7. Participants must be anticipated to be able to complete E-SYNC T cell infusion within 12 weeks after completion of EBRT. 8. All participants must be off systemic steroids for 3 days or more prior to leukapheresis. 9. Must be willing to provide voluntary informed consent for apheresis (and tissue screening if needed) and for study treatment. NOTE: There are two sets of eligibility criteria for Cohort 2. Step 1 defines eligibility for tissue screening and apheresis, and Step 2 defines eligibility for study enrollment and E-SYNC T cell treatment. * Inclusion Criteria for Cohort 2, Step 1: 1. Age \>=18 years. 2. KPS score \>=70. 3. All participants must have adequate organ function defined as: 1. Peripheral absolute neutrophil count \>=1000/mm\^3. 2. Platelet count \>=100,000/mm\^3 (transfusion independent, defined as not receiving platelet transfusions for at least 7 days prior to enrollment). 3. Absolute lymphocyte count (ALC) \>= 300/μL and/or CD3 count of \>=150/μL. 4. Creatinine clearance or radioisotope glomerular filtration rate \>= 50 mL/min/1.73m\^2. 5. Total Bilirubin \<= 1.5 x ULN except for Gilbert's syndrome and 6. Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<= 3x upper limit of normal (ULN). 7. Left ventricular ejection fraction (LVEF) \>= 50% by echocardiogram or multi-gated acquisition scanning (MUGA). 8. Adequate pulmonary function, defined as no evidence of dyspnea at rest and pulse oximetry \> 92% while breathing room air. 4. Pathological criteria: EGFRvIII+ GBM from most recent surgery, as defined by an EGFRvIII + H-score of \>=250 based on central review. 5. All participants must be off systemic steroids for 3 days or more prior to leukapheresis. 6. Must be willing to provide voluntary informed consent for apheresis (and tissue screening if needed). * Inclusion Criteria for Cohort 2, Step 2. Note: Prior to Step 2, participants must have undergone leukapheresis in Step 1. In addition: 1. KPS score \>=70. 2. Must have received at least SOC EBRT as initial therapy; any number of prior recurrences is allowed. 3. Pathological criteria: EGFRvIII+ GBM from most recent surgery, as defined by an EGFRvIII + H-score of \>=250 based on central review. 4. Must have radiographic progression consistent with the Response assessment in neuro-oncology criteria (RANO) criteria for progressive disease (PD) 5. Recurrence must be surgically amenable, with expectation for ability to resect at least 500mg of tumor tissue 6. Participants with reproductive potential agree to use reliable and double barrier method of contraception during the study and for at least 6 months after the last study intervention, including refraining from donating sperm during this period. 7. Females of childbearing potential must have a negative serum beta-Human Chorionic Gonadotropin (hCG) pregnancy test prior to receiving study interventions. 8. All participants must have adequate organ function defined as: 1. Peripheral absolute neutrophil count \>=1000/mm\^3. 2. Platelet count \>=100,000/mm\^3 (transfusion independent, defined as not receiving platelet transfusions for at least 7 days prior to enrollment). 3. Absolute lymphocyte count (ALC) \>= 300/μL and/or CD3 count of \>=150/μL. 4. Creatinine clearance or radioisotope glomerular filtration rate \>= 50 mL/min/1.73m\^2. 5. Total Bilirubin \<= 1.5 x ULN except for Gilbert's syndrome and 6. Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<= 3x upper limit of normal (ULN). 7. Coagulation tests prothrombin time (PT) and partial thromboplastin time (PTT) have to be within normal limits, unless the participant has been therapeutically anticoagulated for previous venous thrombosis. 8. Adequate pulmonary function, defined as no evidence of dyspnea at rest and pulse oximetry \> 92% while breathing room air. 9. Adequate cardiac function, confirmed within the last 12 months, defined as left ventricular ejection fraction (LVEF) \>= 40% by echocardiogram or multi-gated acquisition scanning (MUGA). 9. Must be willing to provide voluntary informed consent for apheresis (and tissue screening if needed). Exclusion Criteria: * Exclusion Criteria for Cohort 1 1. Participant who has been treated with any investigational agents and chemotherapy targeting GBM \<= 4 weeks prior to date of study registration. Exceptions to this include: must be \>=23 days from last dose of temozolomide (TMZ) or radiotherapy, mush be \>= 6 weeks from last dose of nitrosourea. 2. Female participants of reproductive potential who are pregnant or lactating. Female study participants of reproductive potential must have a negative serum pregnancy test as part of eligibility confirmation. 3. Known addiction to alcohol or illicit drugs. 4. Prior treatment with any Epithelial Growth Factor Receptor (EGFR)-targeting therapy. 5. Participants with leptomeningeal dissemination. 6. Participants with a known disorder that affects their immune system, such as human immunodeficiency virus (HIV) or an autoimmune disorder requiring systemic cytotoxic or immunosuppressive therapy. Participants who are currently using non-systemic steroids (e.g., inhaled, intranasal, ocular, topical) are not excluded from the study. 7. Participants with serologic status reflecting active hepatitis B or C infection. Participants that are positive for hepatitis B surface antigen (HBsAg+) will be excluded. Participants that are positive for hepatitis B core antibody or hepatitis C antibody must have a negative polymerase chain reaction (PCR) prior to enrollment. PCR positive participants will be excluded. 8. Participants who have received prior solid organ or bone marrow transplantation. 9. Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, second cancer currently receiving active treatment or anticipated to receive treatment within the next year, or psychiatric illness/social situations that would limit compliance with study requirements. 10. Participants who are unable to return for follow-up visits or obtain follow-up studies required to assess toxicity to therapy. * Exclusion Criteria for Step 1 for Cohort 2 1. Participant who has been treated with any investigational agents or chemotherapy targeting GBM \<=4 weeks prior to date of study registration. Exceptions to this include: must be\>= 23 days from last dose of TMZ or radiotherapy, must be\>=6 weeks from last dose of nitrosourea. 2. Female participants of reproductive potential who are pregnant or lactating. Female study participants of reproductive potential must have a negative serum pregnancy test as part of Step 1 eligibility confirmation. 3. Uncontrolled active infection. 4. Participants with a known disorder that affects their immune system, such as HIV or an autoimmune disorder requiring systemic cytotoxic or immunosuppressive therapy. Participants who are currently using non-systemic steroids (e.g., inhaled, intranasal, ocular, topical) are not excluded from the study. 5. Participants with serologic status reflecting active hepatitis B or C infection. Participants that are positive for hepatitis B surface antigen (HBsAg+) will be excluded. Participants that are positive for hepatitis B core antibody or hepatitis C antibody must have a negative polymerase chain reaction (PCR) prior to enrollment. PCR positive participants will be excluded. 6. Known addiction to alcohol or illicit drugs. * Exclusion Criteria for Step 2 for Cohort 2 Prior to Step 2, participants must have undergone leukapheresis in Step 1. In addition: 1. Prior treatment with any EGFR-targeting therapy 2. Participant who has been treated with any investigational agents or chemotherapy targeting GBM \<= 4 weeks prior to date of study registration. Exceptions to this include: must be \>= 23 days from last dose of TMZ or radiotherapy, must be \>= 6 weeks from last dose of nitrosourea. 3. Participants with imaging or clinical evidence of uncontrolled tumor mass effect; the assessment of mass effect will be made by the Investigators. 4. Participants with leptomeningeal dissemination. 5. Participants with a known disorder that affects their immune system, such as HIV or an autoimmune disorder requiring systemic cytotoxic or immunosuppressive therapy. Participants who are currently using inhaled, intranasal, ocular, topical, or other non-oral or non-IV steroids are not necessarily excluded from the study. 6. Participants with serologic status reflecting active hepatitis B or C infection. Participants that are positive for hepatitis B surface antigen (HBsAg+) will be excluded. Participants that are positive for hepatitis B core antibody or hepatitis C antibody must have a negative polymerase chain reaction (PCR) prior to enrollment. PCR positive participants will be excluded. 7. Participants who have received prior solid organ or bone marrow transplantation. 8. Female participants who are pregnant or breast-feeding. 9. Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, second cancer currently receiving active treatment or anticipated to receive treatment within the next year, or psychiatric illness/social situations that would limit compliance with study requirements. 10. Participants who are unable to return for follow-up visits or obtain follow-up studies required to assess toxicity to therapy.