Osimertinib and Gefitinib in EGFR Inhibitor naïve Advanced EGFR Mutant Lung Cancer

Inclusion Criteria: * Participants must have histologically confirmed stage IV NSCLC (per AJCC 7th edition) with either the L858R or exon 19 deletion activating EGFR mutation as identified in a CLIA-approved laboratory. Note: recurrent stage IV disease initially diagnosed at an earlier stage is considered eligible, provided prior treatment criteria is met. * Participants 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, MRI, or calipers by clinical exam. * Participants can have no prior history of any EGFR-directed therapy, including TKIs or antibodies, and must also be chemotherapy and immunotherapy naïve. Patients who have completed adjuvant or neo-adjuvant chemotherapy or immunotherapy \>6 months ago are considered treatment naive. * Participants must be aged ≥ 18 years * Participants must have an ECOG performance status of 0-1 (Appendix A) * Participants must have a life expectancy of greater than 12 weeks * Participants must have normal organ and marrow function as defined below: * leukocytes ≥3,000/mcL * absolute neutrophil count ≥1,500/mcL * platelets ≥100,000/mcL * hemoglobin \>9.0 g/dL * total bilirubin \< 1.5 times the ULN if no liver metastases or \< 3 times the ULN in the presence of documented Gilbert's syndrome (unconjugated hyperbilirubinemia) or liver metastases * AST(SGOT)/ALT(SGPT) \<2.5 × institutional upper limit of normal or \<5 times the ULN in the presence of liver metastases * creatinine within normal institutional limits OR * creatinine clearance ≥50 mL/min/1.73 m2 for participants with creatinine levels above institutional normal. * Participants should have biopsy tissue at time of diagnosis available for next-generation sequencing testing at the Dana-Farber Cancer Institute. Biopsy can be performed at an outside institution as long as sufficient tissue is available. If next generation sequencing has already been performed prior to study enrollment it does not need to be repeated. Note: Cytology specimen may be acceptable for baseline NGS if tumor cellular content is sufficient and following PI approval. If there is no cytology specimen or tissue sample available for NGS, plasma-based NGS may be acceptable for enrollment following discussion with PI. * Participants must be ≥4 weeks since any major surgery (excluding vascular access placement, mediastinoscopy, or biopsies performed by an interventional service) * Participants must be ≥2 weeks since any prior radiation, including CNS radiation * Male patients: Willing to use adequate contraception (barrier or abstinence) while on treatment with study drug and for 3 months after finishing treatment. * Female patients: Willing to use adequate contraception (barrier or abstinence) while on treatment with study drug and for 3 months after finishing treatment. * Female patients: Must not be pregnant or breast-feeding. Women of child-bearing potential must have a negative pregnancy test prior to start of dosing or must have evidence of non-child-bearing potential by fulfilling one of the following criteria at screening: * Post-menopausal defined as aged more than 50 years and amenorrheic for at least 12 months following cessation of all exogenous hormonal treatments * Women under 50 years are considered postmenopausal if they have been amenorrheic for 12 months or more following cessation of exogenous hormonal treatments and with LH and FSH levels in the post-menopausal range for the institution. * Documentation of irreversible surgical sterilization by hysterectomy, bilateral oophorectomy, or bilateral salpingectomy but not tubal ligation. * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Prior or ongoing treatment with any of the following: * EGFR targeted therapy (TKI or antibody) or any other targeted therapies targeting the ERBB family * Any cytotoxic chemotherapy, investigational agents, or anticancer drugs for the treatment of metastatic NSCLC * Prior radiotherapy within 2 weeks of the first dose of study treatment. Patients who have received radiation to more than 25% of the bone marrow are not eligible at any time. * No uncontrolled central nervous system (CNS) disease, including parenchymal brain metastases, leptomeningeal disease, or spinal cord compression. Patients with asymptomatic untreated brain metastases are eligible. Patients with treated CNS disease will be allowed to enroll provided they have clinically confirmed stable disease with ≥2 weeks since definitive CNS therapy (radiation or surgery) and ≥2 weeks without systemic steroids. Patients may undergo either whole brain radiation or stereotactic radiosurgery prior to study entry. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to gefitinib or osimertinib. * Patients currently receiving (or unable to stop use prior to receiving the first dose of study treatment) medications or herbal supplements known to be potent inhibitors of CYP3A4 (Appendix B). All patients must try to avoid concomitant use of any medications, herbal supplements and/or ingestion of foods with known inducer effects on CYP3A4. The full list of medications that would make a patient ineligible are provided in Appendix B. * Any unresolved toxicities from prior therapy greater than Common Terminology Criteria for Adverse Events (CTCAE) grade 1 at the time of starting study treatment. * Malignancies within the past 3 years excluding adequately treated basal or squamous cell carcinomas of the skin without local or distant metastases. * Refractory nausea and vomiting, chronic gastrointestinal diseases, previous significant bowel resection, or any process that compromises the ability to swallow or absorb oral medication * Significant medical history or unstable medical comorbidities, including: * heart disease including congestive heart failure (NYHA Grade II or greater); unstable angina; prior myocardial infarction (NSTEMI or STEMI) within 6 months prior to study enrollment; hypertension with a systolic blood pressure of \>150 mm Hg or diastolic blood pressure of \>100 mm Hg while on antihypertensive medication * any clinically important abnormalities in rhythm, conduction or morphology of resting ECG, e.g. complete left bundle branch block, third-degree heart block, second-degree heart block, PR interval \>250msec, mean resting corrected QT value (QTc) of \>470msec * any factors that increase the risk of QTc prolongation or risk of arrhythmic events such as heart failure, hypokalemia, congenital long QT syndrome, family history of long QT syndrome or unexplained sudden death under 40 years of age in first degree relatives, or any concomitant medication known to the prolong the QT interval * past medical history of interstitial lung disease, drug-induced interstitial lung disease, radiation pneumonitis which required steroid treatment, or any evidence of clinically active interstitial lung disease * active bleeding diatheses, which in the investigator's opinion makes it undesirable for the patient to participate in the trial or which would jeopardize compliance with the protocol * active infection or ongoing antiviral medication for viral infections including hepatitis B, hepatitis C, or human immunodeficiency virus (HIV). Screening for chronic conditions is not required. HIV-positive participants on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with gefitinib or osimertinib. * ongoing use of warfarin (injectable low-molecular weight heparins are permitted). Patients must be off warfarin for \>7 days prior to enrollment * Identification of EGFR T790M in baseline cfDNA analysis * Active pregnancy or breast-feeding. * Pregnant women are excluded from this study because the effects of gefitinib and osimertinib on the development of the fetus are unknown, and there is potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with gefitinib or osimertinib, breastfeeding should be discontinued if the mother is treated with these agents