Pembrolizumab, Dabrafenib, and Trametinib Before Surgery for the Treatment of BRAF-Mutated Anaplastic Thyroid Cancer

Inclusion Criteria: * Pathologic findings supporting the clinical impression of anaplastic thyroid carcinoma. Diagnosis may include consistent with or suggestive of terminology associated with: anaplastic thyroid carcinoma, undifferentiated carcinoma, squamous carcinoma; carcinoma with spindled, giant cell, or epithelial features; poorly differentiated carcinoma with pleomorphism, extensive necrosis with tumor cells present * Must have a BRAFV600E mutation-positive tumor, as determined by BRAF V600E immunohistochemistry on tumor tissue, genetic/molecular testing of tumor, or cell free (cf)NDA liquid biopsy * Have measurable disease based on RECIST 1.1 * Total bilirubin =\< 1.5 x upper limit of normal (ULN). Total bilirubin =\< 3 x ULN for patients with Gilbert's syndrome * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x ULN, (5 x ULN for patients with concurrent liver metastases) * Serum creatinine =\< within 1.5 x ULN * Absolute neutrophil count (ANC) \>= 1.0 x 10\^9/L * Platelets \>= 100 x 10\^9/L * Hemoglobin \>= 9.0 g/dL or 5.6 mmol/L * International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 x ULN unless participant is receiving anticoagulant therapy as long as PT or activated partial thromboplastin time (aPTT) is within therapeutic range of intended use of anticoagulant * Subjects must be willing to ultimately undergo surgery if their tumor becomes surgically resectable. For MD Anderson site only, subjects must be willing to undergo tumor biopsy after the run-in with DT, unless in the opinion of the treating physician, a biopsy is not feasible or safe. Research subjects retain the right to refuse any research interventions. * Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1. Evaluation of ECOG is to be performed within 7 days prior to the date of allocation/randomization * The participant (or legally acceptable representative if applicable) provides written informed consent for the trial * A male participant must agree to use a contraception of this protocol during the treatment period and for at least 8 months after the last dose of study treatment and refrain from donating sperm during this period * A female participant is eligible to participate if she is not pregnant, not breastfeeding, and at least one of the following conditions applies: * Not a woman of childbearing potential (WOCBP) OR * A WOCBP who agrees to follow the contraceptive guidance during the treatment period and for at least 6 months after the last dose of study treatment Exclusion Criteria: * Significant cardiovascular impairment: history of congestive heart failure greater than New York Heart Association (NYHA) class II * Untreated brain metastases * Prior chemotherapy within \< 1 week prior to study day 1 or patients who have not recovered (i.e., =\< grade 2) from adverse events due to a previously administered agent, except for patients who have been on dabrafenib/trametinib (DT) according to the standard run-in outlined in the trial schema * Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment * History of human immunodeficiency virus (HIV) or active hepatitis B (chronic or acute) or hepatitis C infection. Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen \[HBsAg\] test and a positive anti-HBc \[antibody to hepatitis B core antigen\] antibody test) are eligible. However, patients with past or resolved hepatitis B virus (HBV) should be monitored for reactivation by a specialist. Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA). Note: no testing for HIV, hepatitis B and hepatitis C is required unless mandated by local heath authority * Has a diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of study drug * Has received a live or live-attenuated vaccine within 30 days prior to the first dose of study drug. Administration of killed vaccines is allowed * Has severe hypersensitivity (\>= grade 3) to pembrolizumab and/or any of its excipients * Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial * Females who are breastfeeding or pregnant at screening or baseline (as documented by a positive beta-human chorionic gonadotropin \[beta-hCG\] (or human chorionic gonadotropin \[hCG\]) test with a minimum sensitivity of 25 IU/L or equivalent units of beta-hCG \[or hCG\]). A women of childbirth potential (WOCBP) who has a positive urine pregnancy test within 72 hours prior to the first infusion will be excluded. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * More than 42 days of DT therapy prior to enrollment * A known history of retinal vein occlusion (RVO), central serous retinopathy (CSR), uncontrolled glaucoma or ocular hypertension