Trial of Stereotactic HYpofractionateD RadioAblative (HYDRA) Treatment of Laryngeal Cancer

Study Groups: If participant is found to be eligible to take part in this study, participant will be assigned to a radiation level of HYDRA radiation based on when participant joins this study. Up to 5 dose levels of HYDRA radiation will be tested. Up to 10 participants will be enrolled at each dose level. The first group of participants will receive the lowest dose level. Each new group will receive a higher dose than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable dose of HYDRA radiation is found. HYDRA Radiation Administration: If participant is found to be eligible to take part in this study, participant will receive HYDRA radiation on up to 5 days over the course of about 2 weeks. During radiation, participant will be asked to lie still for about 30-45 minutes. To help make sure that participant's head and neck do not move, participant will wear a head and neck mask. Participant will then have CT or PET-CT scans to make sure that participant is in the right position before participant begins HYDRA radiation. The actual HYDRA radiation treatment will take about 5-10 minutes to complete. Length of Study Participation: Participant may continue receiving HYDRA radiation for up to 2 weeks and for a total of 5 times. Participant will no longer be able to receive the study radiation if the disease gets worse, if intolerable side effects occur, or if participant is unable to follow study directions. Participation on this study will be over after Long-Term Follow-Up. Long-Term Follow-Up may take up to 10 years. Study Visits: On Days that participant receives HYDRA: * Participant will have a speech pathology exam to check participant's voice, swallowing, and breathing function. * Participant will complete the same 5 questionnaires that participant did at screening. Follow-Up Visits: About 6 weeks after participant's last dose of HYDRA radiation: * Participant will have a physical exam. * Participant will have a videostroboscopy to check the status of the disease. * Participant will have a speech pathology exam. * Participant will have a CT or PET-CT scan to check the status of the disease. * Participant will complete the same 5 questionnaires that participant did at screening. * If the doctor thinks it is needed, participant will have a laryngoscopy to check the status of the disease. A laryngoscopy is a standard procedure in which a tube with a lighted camera is inserted through participant's mouth and into participant's throat. * If the doctor thinks it is needed, participant will have a core biopsy to check the status of the disease. To perform a core biopsy, a sample of tissue is removed using a hollow core needle that has a cutting edge. About 3 months after participant's last dose of HYDRA radiation: * Participant will have a physical exam. * Participant will have a modified barium swallow to test participant's swallowing function and to check the status of the disease. * Participant will have a CT or PET-CT scan to check the status of the disease. * Participant will complete the same 5 questionnaires that participant did at screening. * If the doctor thinks it is needed, participant will have a laryngoscopy to check the status of the disease. * If the doctor thinks it is needed, participant will have a core biopsy to check the status of the disease. About 6 months after participant's last dose of HYDRA radiation: * Participant will have a physical exam. * Participant will have a videostroboscopy to check the status of the disease. * Participant will complete the same 5 questionnaires that participant did at screening. * If the doctor thinks it is needed, participant will have a CT or PET-CT scan to check the status of the disease. * If the doctor thinks it is needed, participant will have a speech pathology exam. * If the doctor thinks it is needed, participant will have a laryngoscopy to check the status of the disease. * If the doctor thinks it is needed, participant will have a core biopsy to check the status of the disease. Long-Term Follow-Up: After participant's 6 month follow-up visit, every 3 months for the first 2 years, and then every 6 months after that for up to 5 years: * Participant will have a physical exam. * Participant will have a videostroboscopy to check the status of the disease. * Participant will have a speech pathology exam. * Participant will have a modified barium swallow to test participant's swallowing function and to check the status of the disease. This will only be done between 3-6 months and then 18-24 months after the first HYDRA dose. * Participant will complete the same 5 questionnaires that participant did at screening. * If the doctor thinks it is needed, participant will have a CT or PET-CT scan to check the status of the disease. * If the doctor thinks it is needed, participant will have a laryngoscopy to check the status of the disease. * If the doctor thinks it is needed, participant will have a core biopsy to check the status of the disease. This is an investigational study. HYDRA radiation is delivered using methods that are not FDA-approved or commercially available. It is currently being used for research purposes only. The study doctor can explain how the radiation is designed to work. Up to 50 participants will be enrolled in this study. All will take part at MD Anderson.