LDE225 + Everolimus in Advanced Gastroesophageal Adenocarcinoma

Study Groups: If participant is found to be eligible to take part in this study, participant will be assigned to a combination dose level of the study drugs based on when participant joins this study. Up to 6 participants will be enrolled at each dose level. The first group of participants will receive the lowest dose level of LDE225. Each new group will receive a higher dose than the group before it, if no intolerable side effects were seen. All participants will receive the same dose level of everolimus. This will continue until the highest tolerable combination dose of LDE225 and everolimus is found. Up to 24 participants will be enrolled in this part of the study. Once the highest tolerable dose of the combination of LDE225 and everolimus is found, up to 18 participants will receive the combination dose level that was found to be well tolerated in the earlier part of the study. Study Drug Administration: Every study cycle is 28 days. Participant will take up to 4 capsules or tablets of LDE225 by mouth 1 time every day. Participant's doctor will tell participant how many capsules/tablets participant will need to take. Participant should take the study drug each time with a glass of water (about 8 ounces total) at the same time every day. Participant should swallow the study drug whole, one capsule/tablet every 2 minutes, without chewing or opening the study drug. Participant should eat a light breakfast (such as toast and jam) at least 2 hours before taking the study drug, then participant should not eat or drink anything for at least 1 hour after the dose. Participant will take up to 2 tablets of everolimus by mouth 1 time every day with the LDE225 capsules without food. Participant will take the tablets each time with a glass of water at the same time every day. The tablets should be swallowed whole and not chewed or crushed. If participant cannot swallow the everolimus tablets, the tablets should be dissolved in water just before taking them. Add about 2 tablespoons of water into a glass. The tablets should then be added and stirred gently until the tablets are dissolved. This mixture should then be drunk. Afterwards, the glass should be rinsed with an additional 2 tablespoons of water and drunk. Participant's doctor will discuss this with participant. If participant vomits after taking participant's dose of study drugs, participant should not take another dose that day. It is very important that participant takes the study drugs given to participant just as the study doctor tells participant and that participant does not miss any dose. If participant forgets to take a dose of LDE225, participant should take it as soon as participant remembers (within 6 hours of the missed dose). If more than 6 hours have passed, participant should skip that dose and take participant's next dose the next day as planned. If participant forgets to take a dose of everolimus, participant should skip that dose and take participant's next dose the next day as planned. If participant has a study visit that day, participant should take participant's dose of study drugs at the clinic. Participant will need to store LDE225 in participant's refrigerator at all times. Everolimus can be stored at room temperature. Participant should not share participant's supply of LDE225 and everolimus with anyone. Participant will keep a study drug diary to help participant keep track of participant's study drug doses. If participant misses or delays a dose, participant should write it down in the diary. Participant should bring the diary to every study visit. At the end of each cycle and when participant stops taking part in the study, participant will return any leftover study drugs to the study nurse. Study Visits: On Day 1 of every cycle: * Participant will have a physical exam. * Blood (about 2 teaspoons) will be drawn for routine tests. * If participant can become pregnant, blood (about 1 teaspoon) or urine will be collected for a pregnancy test. Every week during Cycle 1 only, the study staff will call and ask participant how participant is doing. On Day 1 of Cycle 2 and every 2 cycles after that, participant will have an EKG. On Day 1 of Cycle 3 and every 2 cycles after that, participant will have a CT scan or MRI of the chest, abdomen, and pelvis to check the status of the disease. If the doctor thinks it is needed, extra blood may be drawn to follow up on any side effects. Length of Study: Participant may continue taking the study drugs for as long as the doctor thinks it is in participant's best interest. Participant will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, or if participant is unable to follow study directions. Participation on the study will be over after the follow-up visit (described below). End-of-Study Visit: Within 14 days after participant's last dose of study drugs: * Participant will have a physical exam. * Blood (about 2 tablespoons) will be drawn for routine tests, to check the cholesterol levels in participant's blood, and to check how well participant's blood clots. Participant will need to fast for at least 8 hours before this blood draw. * Participant will have an EKG. * Participant will have a CT scan or MRI of the chest, abdomen, and pelvis to check the status of the disease, if the doctor thinks it is needed. * If participant can become pregnant, blood (about 1 teaspoon) or urine will be collected for a pregnancy test. Follow-Up Visit: At 30 days after the end-of-study visit: * Participant will have a physical exam. * Blood (about 2 tablespoons) will be drawn for routine tests. * Participant will have an EKG. * If participant can become pregnant, blood (about 1 teaspoon) or urine will be collected for a pregnancy test. About 3 months after the follow-up visit, participant will be contacted by the study staff by telephone and/or seen in the clinic. Participant will be asked about any drugs participant may be taking and about participant's general health. If participant is called, the call will take about 5-10 minutes. This is an investigational study. LDE225 is not FDA approved or commercially available. Everolimus is FDA approved and commercially available to treat kidney, breast, and pancreatic cancers. The combination of LDE225 and everolimus to treat esophageal or GEJ cancer is investigational. The study doctor can explain how the study drugs are designed to work. Up to 42 participants will be enrolled in this study. All will take part at MD Anderson.