Palbociclib and Cetuximab Versus Cetuximab Monotherapy for Patients With CDKN2A-altered, HPV-unrelated Head and Neck Squamous Cell Carcinoma Who Experienced Disease Progression on a PD-1/L1 Inhibitor

Inclusion Criteria: * Histologically or cytologically confirmed RM-HNSCC that is HPV-unrelated disease; defined as SCC of the oral cavity, larynx, or hypopharynx and p16 negative SCC of the oropharynx or p16 negative non-cutaneous SCC unknown primary of the neck. * CDKN2A loss-of-function (LOF) alteration: mutation or homozygous deletion described on genomic sequencing report. * Measurable disease defined as lesions that can be accurately measured in at least one dimension (longest diameter to be recorded) as ≥ 10 mm with CT scan, as ≥ 20 mm by chest x-ray, or ≥ 10 mm with calipers by clinical exam, per RECIST 1.1. * Disease progression on a PD-1/L1 inhibitor-containing regimen (given as monotherapy or in combination with other therapy). * Received no more than three lines of prior therapy for RM-HNSCC. * At least 18 years of age. * ECOG performance status ≤ 1. * Normal bone marrow and organ function as defined below: * Hemoglobin ≥ 8 g/L * Absolute neutrophil count ≥ 1,000/mcl * Platelets ≥ 100,000/mcl * Total bilirubin ≤ 3 x institutional upper limit of normal (IULN) * AST(SGOT)/ALT(SGPT) ≤ 5 x IULN (for cases involving liver metastases, AST/ALT ≤ 10 x IULN) * Serum creatinine \< 3 x IULN or creatinine clearance \> 30 mL/min by Cockcroft-Gault * The effects of palbociclib and cetuximab on the developing human fetus are unknown. For this reason and because CDK 4/6 inhibitors are known to be teratogenic, women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control, abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while participating in this study, she must inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of the study, and 3 months days after completion of the study * Ability to understand and willingness to sign an IRB approved written informed consent document (or that of legally authorized representative, if applicable). Exclusion Criteria: * Prior treatment with cetuximab for recurrent or metastatic disease (however, prior cetuximab given as a component of multimodality therapy for newly diagnosed, locally advanced, non-metastatic HNSCC is allowable). * Prior treatment with a CDK4/6 inhibitor for RM-HNSCC. * Rb (retinoblastoma) loss: mutation or homozygous deletion described on genomic sequencing report. * Currently receiving any other investigational agents. * A history of other malignancy with the exception of malignancies for which all treatment was completed at least 1 year before registration and the patient has no evidence of recurrent/persistent disease. * Patients with treated brain metastases are eligible if there is no evidence of progression for at least 4 weeks after CNS-directed treatment, as ascertained by clinical examination and brain imaging (MRI or CT scan) during the screening period * A history of allergic reactions attributed to compounds of similar chemical or biologic composition to palbociclib or other agents used in the study (excluding cetuximab). * Prior grade 3 or 4 (per CTCAE 5.0) hypersensitivity reaction to cetuximab. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active serious infection, symptomatic congestive heart failure, unstable angina pectoris, or cardiac arrhythmia. * QTc \>500 msec (using Bazette formula). * Patients with HIV are eligible unless their CD4+ T-cell counts are \< 350 cells/mcL or they have a history of AIDS-defining opportunistic infection within the 12 months prior to registration. Concurrent treatment with effective ART according to DHHS treatment guidelines is recommended. Recommend exclusion of specific ART agents based on predicted drug-drug interactions (i.e. for sensitive CYP3A4 substrates, concurrent strong CYP3A4 inhibitors (ritonavir and cobicistat) or inducers (efavirenz) should be contraindicated).