Brief Summary: This clinical trial tests whether intensity modulated proton therapy after surgery works to shrink tumors in patients with head and neck cancer. Radiation therapy uses high energy protons to kill tumor cells and shrink tumors.

Detailed Description: PRIMARY OBJECTIVE: I. The primary goal is to evaluate the local-regional control among subjects in both arms at 2 years after study registration. SECONDARY OBJECTIVES: I. To determine overall survival progression free survival local, regional, distant recurrence risks, and infield and outfield recurrence in the trial at 2 years after study registration. II. To determine the rate and duration of grade 3+ acute adverse events from treatment start to 30 days after radiation completion date). III. To determine the incidence of secondary acute effects attributable to radiotherapy (e.g. percutaneous endoscopic gastrostomy \[PEG\] tube placement, duration and dose of narcotic analgesia required, weight loss, and hospitalization days). IV. To determine the impact of treatment on patient reported quality of life. V. To objectively quantify the severity of oral mucositis during and following radiotherapy. EXPLORATORY OBJECTIVES: I. To estimate direct and indirect costs of the study regimen and compare these with standard of care treatment techniques. II. To correlate histopathologic, molecular, and tumor genetic/epigenetic alterations with clinical outcomes. III. To correlate circulating biomarkers (microribonucleic acid \[miRNA\], circulating tumor deoxyribonucleic acid \[ctDNA\]) with clinical outcomes. IV. To determine adverse events and patient reported outcomes related to abbreviated concomitant chemotherapy. V. To determine incidence and severity of late effects attributable to radiotherapy at 1-3 years after treatment. VI. To qualitatively evaluate patient beliefs regarding tradeoffs of cancer control, treatment time, cost, acute side effects, and late side effects. VII. To determine cancer out comes, adverse events, and patient reported outcomes and compare across head and neck subsites, between those aged 65 to those age \< 65 at date of enrollment, between male and female, and in the adjuvant population between time to total package completion (\< 9 weeks vs weeks, surgery day 0) and by treatment with and without chemotherapy. VIII. To evaluate the predictive relationship of linear energy transfer (LET) weighted modeling using an relative biologic enhancement (RBE) based model and RBE independent model with grade 3+ acute and late toxicity. OUTLINE: Patients are assigned to 1 of 2 arms. ARM A: Patients who already underwent surgical resection undergo intensity modulated proton therapy (IMPT) for 18 sessions (Monday-Friday) over 24 days in the absence of disease progression or unacceptable toxicity. Patients may receive cisplatin intravenously (IV) over 1-2 hours per standard of care. Patients undergo computed tomography (CT) or magnetic resonance imaging (MRI) and positron emission tomography (PET)/CT or PET/MRI during screening and follow-up. Patients may optionally undergo blood sample collection throughout the trial. ARM B: Patients undergo surgical resection and then IMPT for 15 sessions (Monday-Friday) over 19 days in the absence of disease progression or unacceptable toxicity. Patients may receive cisplatin IV over 1-2 hours per standard of care. Patients undergo CT or MRI and PET/CT or PET/MRI during screening and follow-up. Patients may optionally undergo blood sample collection throughout the trial. After completion of study treatment, patients are followed up within 21 days, every 3 months for 2 years, and then every 6-12 months for 5-10 years.