This study will be testing a combination of two drugs, dasatinib and osimertinib, to overcome resistance to EGFR-TKIs. Osimertinib (AZD9291) is a third-generation EGFR-TKI, which selectively blocks the activity of EGFR mutants, but spares that of wild type. The advantage of using osimertinib is that it inhibits not only the sensitizing EGFR mutations, but also the T790M mutant, which is the most common mechanism of acquired resistance. Dasatinib is a potent, orally available ABL1/SRC TKI, approved for the treatment of chronic myeloid leukemia (CML) in first-line and in patients with imatinib-resistant disease or intolerant, and is being actively studied in patients with advanced solid tumors.