Dabrafenib With or Without Trametinib in Treating Patients With Advanced Differentiated Thyroid Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed papillary thyroid cancer, follicular thyroid cancer (tall cell variant, insular thyroid cancer, follicular variant of papillary thyroid cancers, poorly differentiated thyroid cancer or any of the above mixed histology will be allowed) * Presence of BRAF mutation in tumor tissue * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam; malignant lymph nodes will be considered measurable if they are \>= 15 mm in short axis * Patients must have progressive disease within the thirteen months prior to study entry; progressive disease is as defined in Response Evaluation Criteria in Solid Tumors (RECIST) 1.1, which is at least a 20% increase in the sum of diameters of target lesions and the sum must also demonstrate an absolute increase of at least 5 mm; the appearance of one or more new lesions is also considered progressive disease * Patients are willing to have tumor biopsy pre-study and at 4 weeks on study (fine needle aspiration or core biopsy) if patient has biopsy-accessible tumors as determined by investigator * Patients must have disease that is refractory (unresponsive)…