Osimertinib, Cetuximab, and Tucatinib for the Treatment of EGFR-Mutant Stage IV or Recurrent Non-small Lung Cell Cancer

Inclusion Criteria:

* Participants 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, Exon 20 S768I.
* For patients in the dose escalation cohort, patients must have progressed on any prior EGFR-Tyrosine Kinase Inhibitor (TKI) and if T790M positive must have also progressed on osimertinib. Participants in the dose expansion cohort must have had progressive disease as assessed by Response Evaluation Criteria In Solid Tumors (RECIST) version (v)1.1 criteria on osimertinib as first-line therapy.
* 4\. Participants in the dose expansion cohort must be negative for clinically significant non-ERBB bypass tract resistance such as MET amplification, BRAF mutation, RAS-RAF-MAPK alteration, ALK fusion, RET fusion by a Clinical Laboratory Improvement Amendments (CLIA) (performed after progression on prior EGFR-TKI; if any questions please consult with the Principal Investigator). A defined subset of patients in the expansion cohort will have human epidermal growth factor receptor (HER2) aberrant signaling as putative resistance to osimertinib with overexpression/amplification (or more rarely HER2 Exon 20 insertion mutation) as determined by IHC (3+ IHC) or FISH (HER2/CEP17 \> 2.2 or HER2 copy number \> 6) or NGS for HER2 mutation.
* Adequate archival tissue from a biopsy performed after progression of disease on previous osimertinib (or last EGFR-TKI for dose escalation cohort) or willingness to consent for a fresh tumor biopsy.
* Participants must have measurable disease by RECIST 1.1, by positron emission tomography/computed tomography (PET/CT) or CT imaging within 28 days prior to registration. The CT from a combined PET/CT may be used only if it is of diagnostic quality.
* Men and women \>=18 years of age at the time of consent.
* Eastern Cooperative Oncology Group (ECOG) performance status 0-1.
* Life expectancy greater than 3 months.
* Absolute neutrophil count \>= 1.5 x 10\^9/L.
* Platelets \>= 100 x 10\^9/L.
* Hemoglobin \>= 9 g/dL.
* Serum albumin \>= 2.5 g/dL.
* Bilirubin =\< 1.5 x the upper limit of normal (ULN).
* Alkaline phosphatase (ALP), alanine aminotransferase (ALT), and aspartate transaminase (AST) =\< 2.5 x ULN.
* Potassium within institutional normal limits.
* Magnesium within institutional normal limits.
* Coagulation parameters of international normalized ratio (INR) \< 1.5 and partial thromboplastin time (PTT) \< 30 seconds.
* Creatinine =\< 1.5 x ULN or calculated creatinine clearance \>= 50 mL/min (per the Cockcroft-Gault formula).
* If brain metastases are present, they must be stable and asymptomatic. Participants on corticosteroids for brain metastases are excluded.
* Left ventricular ejection fraction 50% or more on echocardiogram or multigated acquisition scan.
* Individuals of child-bearing potential must test negative for pregnancy =\< 14 days prior to receiving first dose of study medication based on a serum pregnancy test and agree to use 2 methods of birth control or abstain from heterosexual activity during the study and for 6 months following the last dose of the study drug(s), whichever is longer; or be of non- childbearing potential. Non-childbearing potential is defined as (by other than medical reasons) \* \>= 45 years of age and has not had menses for greater than 2 years. \* Amenorrheic for \< 2 years without a hysterectomy and oophorectomy and a follicle-stimulating hormone value in the postmenopausal range upon pretrial (screening) evaluation. \*Post hysterectomy, oophorectomy or tubal ligation. Documented hysterectomy or oophorectomy must be confirmed (of the actual procedure or confirmed by an ultrasound). Tubal ligation must be confirmed (of the actual procedure) otherwise the participant must be willing to use 2 adequate barrier methods throughout the study, starting with the screening visit through 6 months after the last dose of study therapy.
* Ability to swallow pills.
* Ability to understand and willingness to sign an informed consent form.
* Ability to adhere to the study visit schedule and protocol requirements.
* Participants previously on osimertinib allowed to continue prior to initiating study.

Exclusion Criteria:

* History of pneumonitis requiring steroids, including history of radiation-induced pneumonitis.
* Known or suspected leptomeningeal disease as documented by the investigator. Stable asymptomatic brain metastases not requiring steroids is allowed; radiation for brain metastases is allowed as long as there is a two-week washout period prior to starting study treatment.
* Have poorly controlled (\> 1/week) generalized or complex partial seizures, or manifest neurologic progression due to brain metastases notwithstanding central nervous system (CNS)-directed therapy.
* The participant has a known allergy / history of hypersensitivity reaction to any of the treatment components (including any ingredient or similar ingredient used in the product formulations) or red meat.
* History of tick bite or any other contraindication to one of the administered treatments.
* History of arterial or venous thromboembolism within 3 months prior to study enrollment. Participants with a history of venous thromboembolism beyond 3 months prior to study enrollment can be enrolled if they are appropriately treated with low molecular weight heparin or direct oral anti-coagulants.
* History or evidence of current clinically relevant coronary artery disease \>= Grade III by the Canadian Cardiovascular Society Angina Grading Scale or uncontrolled congestive heart failure of current \> Class III as defined by the New York Heart Association.
* The participant has experienced myocardial infarction within 6 months prior to study enrollment.
* The participant has a history of clinically significant ventricular arrhythmia, cardiac arrest, or torsades de pointes.
* The participant has high risk of uncontrolled arrhythmia or uncontrolled cardiac insufficiency.
* The participant has uncontrolled or poorly controlled hypertension (\>180 mmHg systolic or \> 130 mmHg diastolic.
* Use of a strong cytochrome P450 CYP2C8 inhibitor within 5 half-lives of the inhibitor or use of a strong CYP3A4 or CYP2C8 inducer within 5 days prior to the first dose of study treatment.
* The participant is pregnant or breast feeding or expecting to conceive or father children within the projected duration of the trial, starting with the screening visit through 6 months after the last dose of trial treatment.
* The participant has any ongoing or active clinically significant infection, including known active tuberculosis or known ongoing or active infection with the human immunodeficiency virus.
* Recent (within 30 days before enrollment) or concurrent yellow fever vaccination.
* Any condition that would prohibit the understanding or rendering of informed consent.
* Any condition that in the opinion of the investigator would interfere with the participant's safety or compliance while on trial.
* Radiation within 14 days before beginning study drugs.
* Prior EGFR monoclonal antibody treatment (i.e., necitumumab or cetuximab) or HER2 directed treatment (i.e., trastuzumab deruxtecan) for dose expansion only.
* Any systemic therapy within 3 weeks of enrollment with the exception of EGFR-TKI. \* Platinum-based chemotherapy or other lines of intervening systemic therapy after progression on osimertinib or EGFR TKIs is allowed but must be completed no later than 3 weeks before study enrollment.
* Prior EGFR-TKI (if not osimertinib) must be discontinued at least 7 days prior to start of study treatment.