Dabrafenib and Trametinib for Anaplastic Thyroid Cancer

Inclusion Criteria: * Pathologic (histologic or cytologic) diagnosis of anaplastic thyroid cancer (a diagnosis that is noted to be ?consistent with anaplastic thyroid cancer? with the presence of a thyroid mass is acceptable; pathology showing additional types of thyroid cancer is allowed) * Note: Tissue collection for central review is mandatory, but central review is not required for eligibility. Due to the aggressiveness of this disease, treatment will be started prior to central review. * Presence of BRAF mutation (V600E or V600K) in tumor tissue. * Eastern Cooperative Oncology Group (ECOG) performance status \< 2. * Absolute neutrophil count \> 1,000/mcL. * Hemoglobin \>= 9.0 g/dl (Note: The use of transfusion or other intervention to achieve hemoglobin \[Hgb\] \>= 9.0 g/dl is acceptable). * Platelets \> 75,000/mcL. * Total bilirubin \< 1.5 x institutional upper limit of normal (unless due to Gilbert?s disease). * Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) \< 2.5 x institutional upper limit of normal. * Serum creatinine \< 1.5 x institutional upper limit of normal. * Female patients of childbearing potential are required to have a negative serum pregnancy test within 14 days prior to the first dose of study medication. * Females are required to use an effective method of contraception from the time of negative serum pregnancy test, throughout the study duration, and for 4 months after the last dose of study medication. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to study enrollment, for the duration of study participation, and for 16 weeks after completion of the last dose of study drug. * Specific contraception requirements for females: Female subjects of childbearing potential must not become pregnant and are required to be sexually inactive by abstinence or use contraceptive methods with a failure rate of \< 1%. Sexual inactivity by abstinence must be consistent with the preferred and usual lifestyle of the subject. Periodic abstinence (e.g. calendar, ovulation, symptothermal, post ovulation methods) and withdrawal are not acceptable methods of contraception. Contraceptive methods with a failure rate of \< 1% include the following: * Intrauterine device (IUD) or intrauterine system (IUS) that meets the \< 1% failure rate as stated in the product label, * Male partner sterilization (vasectomy with documentation of Azoospermia) prior to the female subject's entry into the study, and this male is patient?s sole sexual partner. For this definition, ?documented? refers to the outcome of the investigator's/qualified physician designee?s medical examination of the subject or review of the subject's medical history for study eligibility, as obtained via a verbal interview with the subject or from the subject?s medical records. * Double barrier method: condom and occlusive cap (diaphragm or cervical/vault caps) plus spermicidal agent (foam, gel, film, cream, suppository) These allowed methods of contraception are only effective when used consistently, correctly and in accordance with the product label. The investigator is responsible for ensuring subjects understand how to properly use these methods of contraception. * Specific contraception requirements for males: To prevent pregnancy in a female partner or to prevent exposure of any partner to the investigational product from a male subject?s semen, male subjects must use one of the following contraceptive methods during the study and for a total of 16 weeks following the last dose of study drug (based upon the lifecycle of sperm): * Abstinence, defined as sexual inactivity consistent with the preferred and usual lifestyle of the subject for 14 days prior to first dose of study drug, through the dosing period, and for at least 16 weeks after the last dose of study drug. Periodic abstinence (e.g. calendar, ovulation, symptothermal, post-ovulation methods) and withdrawal are not acceptable methods of contraception. * Condom (during non-vaginal intercourse with any partner - male or female) OR * Condom and occlusive cap (diaphragm or cervical/vault caps) plus spermicidal agent (foam/gel/film/cream/suppository) (during sexual intercourse with a female). * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Patients with resectable stage IVA anaplastic thyroid cancer who are candidates for surgery and wish to proceed with surgery. * Patients who have had external beam radiotherapy to neck or chest for cancer that would result in overlap of radiation therapy fields. * Patients who have had cytotoxic chemotherapy, stereotactic brain radiation or external beam radiation within 2 weeks prior to study treatment initiation. * Patients who have had oral multikinase inhibitors within 1 week prior to study treatment initiation. * Patients that have not recovered from adverse events related to prior therapy for cancer to Common Terminology Criteria for Adverse Events (CTCAE) 4.03 grade 2 or less except for alopecia. * Patients previously treated with potent BRAF inhibitor or MEK inhibitor. Previous treatment with sorafenib is permitted. * Patients that are receiving any other investigational agent. * Patients that are currently taking any prohibitive medication. * Patients with a history of other active malignancy requiring cancer treatment. * Patients with uncontrolled brain metastases. Patients who are on a stable dose of corticosteroids for more than 1 week or off corticosteroids for 2 weeks prior to study enrollment can be enrolled. Enzyme-inducing anti-epileptic drugs are not permitted. * Patients with a known history of retinal vein occlusion (RVO), central serous retinopathy (CSR), uncontrolled glaucoma or ocular hypertension. * Patients with class II, III, or IV heart failure as defined by the New York Heart Association (NYHA) functional classification system. * Corrected QT (QTc) interval greater than or equal to 480 msecs (\>= 500 msec for subjects with Bundle Branch Block). * Patients with uncontrolled intercurrent illness including, but not limited to, ongoing or active infection requiring intravenous (IV) antibiotics, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant women and nursing women are excluded from this study because dabrafenib has the potential for teratogenic or abortifacient effects. In embroyfetal developmental studies in rats, developmental toxicities including reduced fetal body weight, embryo-lethality, cardiac ventricular septal defect malformations, delayed skeletal development and variation in thymic shape have been observed. * Known human immunodeficiency virus (HIV)-positive patients or those on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with study drugs.