Testing the Combination of MLN0128 (TAK-228) and AZD9291 in Advanced EGFR (Epidermal Growth Factor Receptor) Mutation Positive Non-small Cell Lung Cancer

Inclusion Criteria: * Patients with stage IV or recurrent/metastatic histologically or cytologically confirmed non-squamous NSCLC * NSCLC must harbor an EGFR activating mutation (Exon 21 L858R, Exon 19 deletion) * Progressive disease on osimertinib (AZD9291) given first line * For the dose expansion portion ONLY, patient must: 1) have progression of disease with first line osimertinib administered for advanced or metastatic disease as the last previous systemic treatment, 2) be treatment naïve for other 3rd generation EGFR-TKI (CO-1686) and mTOR inhibitors * Patients must have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1, defined as at least one lesion that can be accurately measured in at least one dimension \>= 10 mm (\>= 1 cm) by computed tomography (CT) imaging or magnetic resonance imaging (MRI) within 28 days prior to start of protocol therapy; the CT from a combined positron emission tomography (PET)/CT may be used if it is of diagnostic quality; laboratory parameters are not acceptable as the only evidence of disease * For the dose expansion, no other systemic therapies for advanced/metastatic disease is permissible after first line osimertinib * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%) * Patients with a prior history of brain metastases are eligible provided: * The brain metastases have been treated; patients with small brain metastases for which radiation or surgery is not indicated…