Brief Summary: This phase I trial studies the best dose of sonidegib when given together with pembrolizumab and to see how well they work in treating patients with solid tumor that has spread to other places in the body (advanced). Sonidegib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Giving sonidegib and pembrolizumab may work better than standard treatment in treating patients with advanced solid tumors.

Detailed Description: PRIMARY OBJECTIVES: I. To determine the maximum tolerated dose (MTD) of sonidegib in combination with pembrolizumab in participants with advanced solid tumors as part of the dose escalation phase. (Part A) II. To estimate the response rate of sonidegib in combination with pembrolizumab in participants with non-small cell lung cancer (NSCLC) or head and neck squamous cell carcinoma (HNSCC) as part of the expansion cohort based on Response Evaluation Criteria in Solid Tumors (RECIST) criteria. (Part B) SECONDARY OBJECTIVES: I. To characterize the safety profile and tolerability of sonidegib and pembrolizumab. II. To obtain preliminary estimates of efficacy as measured by response rate (based on RECIST criteria), disease control rate at 6 months, duration of response, overall survival (OS), and progression free survival (PFS) of sonidegib and pembrolizumab in patients with selected advanced solid tumors. CORRELATIVE RESEARCH OBJECTIVE: I. To estimate the immunologic effects of sonidegib and pembrolizumab by investigating the changes in circulating tumor cells, immune cell markers, cytokines, and soluble PD-L1 in blood. OUTLINE: This is a dose-escalation study of sonidegib. Patients receive sonidegib orally (PO) once daily (QD) on days 1-8 and pembrolizumab intravenously (IV) over 30 minutes on day 8. Treatment repeats every 21 days for up to 24 cycles in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up within 30 days.