A Study of Avutometinib and Defactinib in People With Thyroid Cancer

Inclusion Criteria: Cohort A will enroll RAIR, R/M DTC patients with RAF dimer-driven disease. Cohort B will enroll ATC patients with RAF dimer-driven disease. * Cohort A only: Patients must have pathologically or cytologically confirmed differentiated thyroid cancer of follicular origin (including papillary thyroid carcinoma, follicular thyroid carcinoma, hurthle cell carcinomas, poorly differentiated thyroid carcinoma and their respective variants). * Cohort B only: Patients must have anaplastic thyroid carcinoma. * Confirmation in a CLIA certified laboratory that one of the patient's thyroid tumors (primary tumor, recurrent tumor, or metastases) possess at least one of the following genetic alterations: RAS mutation, NF1 mutation, RET rearrangement, NTRK rearrangement, ALK rearrangement, Class 2 or 3 BRAF alterations (non-V600E/K mutations or rearrangements). Patients with anaplastic thyroid cancer harboring a Class 1 BRAF alteration (V600E/K/D/R) also would be eligible if the patient has previously been treated with BRAF-targeted therapy (for example, dabrafenib plus trametinib, vemurafenib plus cobimetinib, encorafenib plus binimetinib, etc…). * Cohort A only: Evidence of progressive disease (e.g. presence of new or growing lesion(s) on radiologic imaging and/or new or worsening tumor-related symptoms) within 14 months of study enrollment. * Cohort A only: Patients must have recurrent or metastatic disease not amenable to curative surgery or radiation. * Patients with any number of prior therapies will be eligible. * Patients must have RECIST v1.1 measurable disease. * Age ≥ 18 years. * ECOG performance status of 0 or 1. * For Cohort A only: Patients must have not had recent treatment for thyroid cancer as defined as: * No prior RAI therapy is allowed \<6 months prior to initiation of therapy on this protocol. A diagnostic study using \<10 mCi of RAI is not considered RAI therapy * No external beam radiation therapy \<1 weeks prior to initiation of therapy on this protocol. * No chemotherapy or targeted therapy (e.g., tyrosine kinase inhibitor) is allowed \<4 weeks prior to the initiation of therapy on this protocol * For Cohort A only: Patients must have RAI-refractory disease, defined as one of the following: * Total lifetime dose of radioiodine \> 600 mCi * A tumor that is not radioiodine-avid on a diagnostic radioiodine scan performed * A radioiodine-avid metastatic lesion which progressed despite radioiodine treatment given 6 months or more prior to study entry in the study. There are no size limitations for the index lesions used to satisfy this entry criterion * The presence of at least one fluorodeoxyglucose (FDG) avid lesion. * Patients must be able to swallow and retain orally-administered pills without any clinically significant gastrointestinal abnormalities that may alter absorption, such as malabsorption syndrome or major resection of the stomach or bowels. * Adequate recovery from toxicities related to prior treatments to at least Grade 1 by CTCAE v 5.0. Exceptions include alopecia and peripheral neuropathy grade ≤ 2. * Patients must have tissue from the primary tumor or metastases available for correlative studies. Either a paraffin block or at least 20 unstained slides are acceptable (30 unstained slides would be ideal). (If less than twenty unstained slides are available and a paraffin bloc is not available, the patient may be able to participate at the discretion of the investigator). * Patients must agree to undergo two research biopsies of (a) malignant lesion(s). Tumor tissue obtained prior to study consent or treatment as part of standard of care can also be submitted in lieu of performance of the first pre-treatment biopsy if the Principal Investigator deems it to be of sufficient quantity/quality/timeliness. Patients may also be exempt from biopsy if 1) the investigator or person performing the biopsy judges that no tumor is accessible for biopsy, 2) the investigator or person performing the biopsy feels that the biopsy poses too great of a risk to the patient (including if conduct of the biopsy will result in an unacceptable delay in therapy), or 3) the patient cannot be safely removed from anti-coagulation therapy (if the anti-coagulation therapy needs to be temporarily held for the biopsy procedure). If the only tumor accessible for biopsy is also the only lesion that can be used for RECIST v1.1 response evaluation, then the patient may be exempt from biopsy. If the investigator deems a second research biopsy to be high risk after a patient has completed the first research biopsy, the patient may be exempt from the second biopsy. Biopsies of lesions that are in proximity to any vital neurovascular structures that can be considered high risk procedures will not be biopsied. * Baseline QTc interval \< 460 ms for women and ≤450 ms for men using Frederica's QT correction formula. NOTE: This criterion does not apply to patients with a right or left bundle branch block. * Adequate cardiac function wit left ventricular ejection fraction \>50% by echocardiography (ECHO) or multiple-gated acquisition (MUGA) scan. * Screening laboratory values must meet the following criteria: * WBC ≥ 2000/μL * Neutrophils ≥ 1000/μL * Platelets ≥ 100 x10\^3 /μL * Hemoglobin \> 9.0 g/dL * AST/ALT ≤ 2.5 x ULN (of \< 5x ULN in patients with liver metastases) * Total Bilirubin ≤ 1.5 x ULN (except subjects with Gilbert Syndrome, who can have total bilirubin \< 3.0 mg/dL) * International normalized ratio (INR) \< 1.5 and partial thromboplastin time (PTT) \< 1.5 x ULN in the absence of anticoagulation or therapeutic levels in the presence of anticoagulation. * Albumin ≥ 3.0 g/dL (451 μmole/L) * Creatine phosphokinase (CPK) ≤ 2.5 x ULN * Serum creatinine ≤ 1.5 x ULN or creatinine clearance (CrCl) ≥ 50 mL/min (if using the Cockcroft-Gault formula below) * Female CrCl = (140 - age in years) x weight in kg x 0.85 72 x serum creatinine in mg/dL * Mal