Study of the CDK4/6 Inhibitor Palbociclib (PD-0332991) in Combination With the PI3K/mTOR Inhibitor Gedatolisib (PF-05212384) for Patients With Advanced Squamous Cell Lung, Pancreatic, Head & Neck and Other Solid Tumors

Inclusion Criteria: * For Part I, participants must have histologically confirmed malignancy that is metastatic or unresectable and resistant to standard therapy or for which no standard therapy is available. For Part II, participants must have histologically confirmed advanced squamous cell lung cancer, advanced pancreatic cancer, advanced head \& neck cancer (specifically non-oropharynx squamous cell carcinoma or HPV-negative oropharynx squamous cell carcinoma), or any tumor with suspected PI3K-pathway dependence (either by mutation or by known biologic rationale, such as endometrial cancer. PI3K dependence includes the presence of a PIK3CA-mutant hotspot mutation, PIK3CA copy number gain, or PTEN loss in the archival or fresh tumor tissue specimen identified in a CLIA-certified laboratory. All genetic findings must be reviewed by the study PI, prior to study entry.) * For Part I, participants are required to have only evaluable disease (disease that is visible on imaging studies but does not meet RECIST criteria for measurable disease). For Part II, 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) as \> 20 mm with conventional techniques or as \> 10 mm with spiral CT scan. See section 10 for the evaluation of measureable disease. * Participants are permitted to have any number of prior therapies prior to enrollment * Age ≥ 18 years. . * ECOG performance status ≤ 2 * Participants must have normal organ and marrow function as defined below: * Absolute neutrophil count ≥ 1,500/mcL * Hemoglobin ≥9.0 gm/dL * Platelets ≥ 100,000/mcL * Total bilirubin within normal institutional limits * AST (SGOT)/ALT (SGPT) ≤ 2.5 X institutional upper limit of normal (≤ 5.0 X institutional upper limit of normal permitted if hepatic metastases present) * Creatinine within 1.5x the ULN institutional limits * Fasting glucose ≤ 126 mg/dL (7.0 mmol/L) * HbA1c ≤ 7.0% * The effects of palbociclib and Gedatolisib (PF-05212384) on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and 90 days after discontinuation. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Participants who have had chemotherapy or radiotherapy within 3 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from acute effects of any prior therapy to baseline or Grade ≤1 except for Alopecia or AEs not constituting a safety risk in the opinion of the investigator. * Participants may not be receiving any other study agents concurrently with the study drugs. * Participants with symptomatic brain metastases that require active treatment are excluded. * Current use or anticipated need for food or medications that are known strong CYP3A4 inhibitors/inducers, including their administration within 7-days prior to the first Gedatolisib (PF-05212384) or palbocilib dose and during study treatment. * QTc \> 480 msec; history of QT syndrome, Brugada syndrome or known history of QTc prolongation, or Torsade de Pointes. * Patients with a history of diabetes. * Uncontrolled intercurrent illness including, but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant women are excluded from this study because the study agents have the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk of adverse events in nursing infants secondary to treatment of the mother with the study agents, breastfeeding should be discontinued if the mother is treated. * Individuals with a history of other malignancy are ineligible except for the following circumstances. Individuals with a history of previous malignancies are eligible if they have been disease-free for at least 2 years and are deemed by the investigator to be at low risk for recurrence of that malignancy. Individuals with the following cancers are eligible if diagnosed and treated within the past 5 years: cervical cancer in situ, and basal cell or squamous cell carcinoma of the skin. * Active and clinically significant bacterial, fungal or viral infection including hepatitis B (HBV), hepatitis C (HCV), known human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS)-related illness. * Concurrent use or anticipated need for medications that are mainly metabolized by UGT1A9 including their administration within 7-days prior to the first dose of study drug. * Current use or anticipated need for food or medications that are metabolized by CYP2D6, and of narrow therapeutic index including their administration within 10-days prior to the first Gedatolisib (PF-05212384) dose and during study treatment. * Concurrent use of herbal preparations including saw palmetto. * Current use of drugs known to prolong QT interval. * Concurrent use of proton pump inhibitors (including, but not limited to, dexlansoprazole, esomeprazole, lansoprazole, omeprazole, pantoprazole, rabeprazole) with palbociclib is prohibited.