Osimertinib and Necitumumab in Treating Patients With EGFR-Mutant Stage IV or Recurrent Non-small Cell Lung Cancer Who Have Progressed on a Previous EGFR Tyrosine Kinase Inhibitor

Inclusion Criteria: * Patients with stage IV or recurrent/metastatic histologically confirmed non-small cell lung cancer (NSCLC) * NSCLC must harbor at least one of the following EGFR activating mutations: Exon 21 L858R, Exon 19 deletion, Exon 18 G719X, Exon 21 L861Q or for EGFR Exon 20 insertion expansion cohort D, NSCLC must harbor an EGFR Exon 20 insertion performed by a Clinical Laboratory Improvement Act (CLIA) certified test * For Dose escalation cohort - progressive disease on at least one prior EGFR-TKI (previous treatment with 3rd generation EGFR-TKI including osimertinib \[AZD9291\] allowed for dose escalation) * For Dose Expansion Cohort A: patient must 1) have progression of disease on erlotinib, gefitinib or afatinib as last previous systemic treatment, 2) have biopsy of tumor taken after progression on erlotinib, gefitinib or afatinib which must be EGFR-T790M negative confirmed by central testing prior to treatment (if EGFR-T790M status is unknown, patients may consent for trial and for biopsy and testing for EGFR T790M will be performed as part of initial biopsy for trial), and 3) be treatment naive to 3rd generation EGFR-TKI (rociletinib, EGFR inhibitor HM61713 \[HM61713\] and osimertinib \[AZD9291\]) and EGFR monoclonal antibodies * For Dose Expansion Cohort B (closed to accrual as of 8/30/18): patient must 1) have progression of disease on a 3rd generation EGFR-TKI such as AZD9291, rociletinib, HM61713, 2) be treatment naive to an EGFR monoclonal antibody, …