Study of Cemiplimab Combined With Dabrafenib and Trametinib in People With Anaplastic Thyroid Cancer

Inclusion Criteria: * Pathological (histologically or cytologically) proven diagnosis of BRAF-V600E mutant ATC (a diagnosis that is noted to be consistent with ATC is acceptable) * Either Metastatic disease or locoregional disease that is considered not resectable for cure * Ideally a surgeon should determine that the disease is not resectable for cure, but this can also be done by any investigator * Patients must have measurable disease according to RECIST 1.1 criteria, defined as at least 1 lesion that can be accurately measured in at least 1 dimension (longest diameter to be recorded for nonnodal lesions and short axis for nodal lesions) as \>/= 20 mm with conventional techniques or as \>/= 10 mm with spiral CT scan, MRI, or calipers by clinical exam * Age \>/= 18 years * Eastern Cooperative Oncology Group (ECOG) performance status \</= (or Karnofsky performance score \>/= 60) * Able to swallow and retain orally administered medication * Patient must have normal organ and marrow function as defined below: * Absolute neutrophil count \>/=1.5 x 10\^9/L * Hemoglobin \>/=8 g/dL * Platelets \>/=100 x 10\^9/L * Serum bilirubin \</=1.5x institutional ULN (unless the patient has GIlbert's Disease, in which case total bilirubin \</=3x institutional ULN) * AST and ALT \</=2.5x institutional ULN (\</=5x institutional ULN if there is liver metastasis) * Serum creatinine \</=1.5mg/dL or calculated creatinined clearance (Cockcroft-Gault formula) \>/=50 mL/min or 24-h urine creatinine clearance \>/=50 mL/min * Left ventricular ejection fraction greater than or equal to instutional lower limit of normal (LLN) by echocardiogram or multigated acquisition (MUGA) * Negative pregnancy test (serum or urine) within 14 days of registration for women of childbearing potential. Women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) before study entry and for the duration of study participation. Men treated or enrolled on this protocol must also agree to use adequate contraception before the study, for the duration of study participation, and for 4 months after completion of trametinib administration * Must agree to allow 2-4 separate biopsies of any malignant lesion. For patients whose biopsies (initial) are deemed as unsafe or contraindicated, they will not be eligible. * Ability to understand and willingness to sign a written informed consent document. Note: Use of Legally Authorized Representative (LAR) is permitted Exclusion Criteria: * Previous documentation or current evidence of treatment with dabrafenib and trametinib. ° Exception: (1) Patients who started dabrafenib and tranetinib for ATC at an institution outside of MSK are eligible or (2) with the consent of the PI (Sherman). However, this exception is limited to 8 subjects. * Active brain metastases, unless an exception is granted by the Principal Investigator. * Current interstitial lung disease or pneumonitis * Prior history of idelalisib therapy. Exceptions allowed with the consent of the principal investigator (Dr. Sherman) * History of retinal vein occlusion (RVO) or central serous retinopathy (CSR): ° History of RVO or CSR or predisposing factors to RVO or CSR (e.g. uncontrolled glaucoma or ocular hypertension) * History or current evidence of cardiovascular risk, including any of the following: * Left ventricular ejection fraction (LVEF) \<LLN * A QT interval corrected for heart rate using the Bazett's formula of QTcB\>/=480msec * Current evidence of clinically significant uncontrolled arrhythmias (exception: patients with controlled atrial fibrillation for \>30 days before enrollment are eligible) * History of acute coronary syndromes (including myocardial infarction and unstable angina), coronary angioplasty, or stenting within 6 months before treatment * Known hepatitis B virus (HBV) or hepatitis C virus (HCV) infection (with the exception of chronic or cleared HBV and HCV infection, which will be allowed) HIV-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with trametinib. In addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy * Uncontrolled intercurrent illness that would limit compliance with study requirement. * Inability to receive immunotherapy for the following reasons: * Any prior grade \>/=3 immune-related adverse event (irAE) while receiving any previous immunotherapy agent or any unresolved irAE grade \>1 * Active or prior documented autoimmune disease within the past 2 years. NOTE: Subjects with vitiligo, Grave's disease, or psoriasis not requiring systemic treatment (within the past 2 years) are not excluded. Exceptions allowed with the consent of the principal investigator (Dr. Sherman) * Active inflammatory bowel disease (e.g., Crohn's disease, ulcerative colitis) * History of primary immunodeficiency * History of allogeneic organ transplant * Known history of previous clinical diagnosis of active tuberculosis (this does not include a history of being PPD positive)