Trametinib in Increasing Tumoral Iodine Incorporation in Patients With Recurrent or Metastatic Thyroid Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed thyroid carcinoma of follicular origin (including papillary, follicular, or poorly differentiated subtypes and their respective variants); confirmation of thyroid carcinoma will be done at Memorial Sloan-Kettering (MSK) * 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 CT scan, magnetic resonance imaging (MRI), or calipers by clinical exam; tumors in previously irradiated fields may be considered measurable if there is evidence of tumor progression after radiation treatment * RAI-refractory disease on structural imaging, defined as any one of the following: * A metastatic lesion that is not radioiodine-avid on a diagnostic radioiodine scan performed prior to enrollment in the current study, or * A radioiodine-avid metastatic lesion which remained stable in size or progressed despite radioiodine treatment 6 months or more prior to entry in the study; there are no size limitations for the index lesion used to satisfy this entry criterion * The presence of at least one fluorodeoxyglucose (FDG) avid lesion * No recent treatment for thyroid cancer as defined as: * No prior RAI therapy is allowed \< 6 months prior to initiation of therapy on this protocol…