Mosunetuzumab With Chemotherapy for the Treatment of Patients With Untreated C-Myc Rearrangement Positive High Grade B Cell Lymphoma or Diffuse Large B Cell Lymphoma

Inclusion Criteria: * For both phases of the study, participant must be 18-75 years of age and have previously untreated high-grade B cell lymphoma (HGBCL) or diffuse large B cell lymphoma (DLBCL), including transformed DLBCL per the World Health Organization (WHO) 2022 classification, and with documented c-Myc rearrangement on fluorescence in situ hybridization (FISH) testing. Eligible types of c-Myc rearrangements will be performed by FISH testing and may include any single MYC rearrangement (single-hit lymphoma \[SHL\]), Double hit (DHL) lymphoma or and triple hit (THL) lymphoma defined by translocations of MYC and BCL2 (DHL) and BCL6 (THL) * Pathology must be verified and confirmed by university pathologists at the enrolling institution and centrally (OHSU) for any biopsies read outside of either institution * Stage II or higher and International Prognostic Index (IPI) score of 2-5 * Able to comply with the study protocol and procedures, in the investigator's judgment * At least one bi-dimensionally measurable nodal lesion, defined as ≥ 1.5 cm in its longest dimension, or one bi-dimensionally measurable extranodal lesion, defined as ≥ 1.0 cm in its longest diameter (patients with measurable disease prior to the pre-phase who have disappearance of measurable disease at initiation of study therapy cycle 1 day 1 \[C1D1\] are eligible) * Confirmed availability of archival or freshly collected tumor tissue before study enrollment * Eastern Cooperative Oncology Group Performance Status of 0, 1, or 2 * Left ventricular ejection fraction (LVEF) defined by multiple-gated acquisition (MUGA) scan or echocardiogram (ECHO) within the institutional limits of normal * Absolute neutrophil count (ANC) ≥ 1.0 ×10\^9/L unless inadequate function is due to underlying disease, as established by extensive bone marrow involvement, or is due to hypersplenism secondary to the involvement of the spleen by lymphoma per the investigator) without transfusion * Platelet count ≥ 75 ×10\^9/L (unless inadequate function is due to underlying disease, as established by extensive bone marrow involvement, or is due to hypersplenism secondary to the involvement of the spleen by lymphoma per the investigator) without transfusion * Serum creatinine ≤ upper limit of normal (ULN); or estimated creatinine clearance ≥ 50 mL/min by Cockcroft Gault method or other institutional standard methods, e.g. based on nuclear medicine renal scan * For persons of childbearing potential (PCBP), an agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods that result in a failure rate of \< 1% per year, and confirmed agreement to refrain from donating eggs, during the treatment period and for at least 3 months after the last dose of mosunetuzumab, and 3 months after the last dose of tocilizumab (if applicable), whichever is longer * For participants who can produce sperm and create pregnancy: confirmed agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agreement to refrain from donating sperm Exclusion Criteria: * Pregnant or breast /chestfeeding * Prior treatment for DLBCL. Exceptions: * Course of bendamustine plus rituximab (BR) treatment \> 3 years prior for follicular lymphoma, or any history of rituximab treatment * As a pre-phase therapy, the following are allowed: * Prednisone of ≤ 100 mg for up to a total of 14 days. Prednisone or equivalent corticosteroid must be discontinued by the time of treatment start. These days do NOT have to be consecutive (i.e., can include multiple courses as long as ≤ 14 days) * One cycle of RCHOP (can be dose reduced) or DA R EPOCH or BR or single agent rituximab * Radiation to up to 3 disease sites * For chemotherapy and radiation, a washout of 3 weeks is required (exception: for mediastinal/pericardial radiation, the washout is 4 weeks) * History of severe allergic or anaphylactic reactions to humanized or murine monoclonal antibodies or known sensitivity or allergy to murine products * Contraindication to receive full dose of any of the individual components of EPOCH * Participants with history of confirmed progressive multifocal leukoencephalopathy (PML) * Known or suspected chronic active Epstein Barr virus (CAEBV) infection * Positive test results for chronic hepatitis B infection (defined as positive hepatitis B surface antigen \[HBsAg\] serology) \* Participants with occult or prior hepatitis B infection (defined as positive total hepatitis B core antibody and negative HBsAg) may be included if hepatitis B virus (HBV) deoxyribonucleic acid (DNA) is undetectable at the time of screening. These Participants must be willing to undergo monthly DNA testing and appropriate antiviral therapy as indicated * Acute or chronic hepatitis C virus (HCV) infection. Participants positive for HCV by antibody testing, but negative for HCV by polymerase chain reaction (PCR) are eligible * HIV seropositivity * Administration of a live, attenuated vaccine within 4 weeks before first study treatment administration or anticipation that such a live, attenuated vaccine will be required during the study * Prior solid organ transplantation * Known or suspected history of hemophagocytic lymphohistiocytosis (HLH). * History of autoimmune disease, including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener granulomatosis, Sjögren syndrome, Guillain-Barré syndrome, multiple sclerosis, vasculitis, or glomerulonephritis. Exceptions: * Participants with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement may be eligible. * Participants with controlled Type 1 diabetes mellitus who are on an insulin regimen are eligible for the study. * Participants with a history of disease-related immune thrombocytopenic purpura, autoimmune hemolytic anemia, or other stable autoimmune diseases may be eligible after review and approval by the primary investigator (PI) * Systemic immunosuppressive medications (including, but not limited to, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor agents) with the exception of pre-phase treatment with prednisone up to 100 mg daily for 7 days (or equivalent corticosteroid dose) prior to cycle 1 day 1 (C1D1). Exceptions: * The use of inhaled corticosteroids is permitted. * The use of mineralocorticoids for management of orthostatic hypotension is permitted. * The use of physiologic doses of corticosteroids for management of adrenal insufficiency is permitted * Known active central nervous system (CNS) involvement of lymphoma * Current or past history of CNS disease, such as stroke, epilepsy, CNS vasculitis, neurodegenerative disease. Exceptions: * Participants with a history of stroke who have not experienced a stroke or transient ischemic attack in the past 2 years and have no residual neurologic deficits as judged by the investigator are allowed. * Participants with a history of epilepsy who have had no seizures in the past 2 years while not receiving any anti-epileptic medications are allowed in the expansion cohorts only * Prior radiotherapy to the mediastinal / pericardial region within 4 weeks * Prior pre-phase chemotherapy and radiation, within 3 weeks is required (Exception noted for mediastinal/pericardial radiation) * Malignancy treated with curative intent unless in documented remission without treatment for 2 years prior to enrollment, or other malignancy that could affect compliance with the protocol or interpretation of results. Exception: Participants with a history of curatively treated basal or squamous cell carcinoma or melanoma of the skin or in situ carcinoma of the cervix are eligible. Adjuvant endocrine therapy for non-metastatic, hormone receptor-positive breast cancer is permitted * Evidence of significant, uncontrolled concomitant diseases that could affect compliance with the protocol or interpretation of results or that could increase risk to the Participant, including renal disease that would preclude chemotherapy administration or pulmonary disease (including obstructive pulmonary disease and history of bronchospasm) * Significant pulmonary disease (including obstructive pulmonary disease and history of bronchospasm) * Significant cardiovascular disease, defined as * New York Heart Association \[NYHA\] Class III or IV cardiac disease, * Congestive heart failure, * Myocardial infarction within the previous 6 months, * Unstable arrhythmias, or * Unstable angina * Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) at study enrollment or any major episode of infection requiring treatment with IV antibiotics or hospitalization (relating to the completion of the course of antibiotics) within 10 days before C1D1 * Clinically significant history of liver disease, including viral or other hepatitis, current alcohol abuse, or cirrhosis * Aspartate aminotransferase (AST) or Alanine aminotransferase (ALT) \> 2.5 x ULN * Total bilirubin ≥ 1.5 x ULN * International normalization ratio (INR) \> 1.5 x ULN in the absence of therapeutic anticoagulation * Partial prothrombin time (PTT) or adjusted partial prothrombin time (aPTT) \> 1.5 x ULN in the absence of a lupus anticoagulant * Herbal therapies intended as treatment of lymphoma * Medicinal or recreational cannabis products are not permitted while receiving the study intervention.