In a medium saucepan, combine the apples, water, maple syrup and cinnamon stick and bring to a boil. Simmer over moderately low heat, stirring occasionally, until the apples are tender, about 6 minutes. Using a slotted spoon, transfer the apples to a bowl and let cool to room temperature. In a shallow bowl, whisk the milk and eggs. Spread 4 slices of the bread with mustard. Top with half of the cheese, the ham and then the remaining cheese. In a large skillet, melt 1 tablespoon of the butter over low heat. Dip the bottoms of the 4 topped bread slices in the beaten egg until just saturated and transfer to the skillet. Dip the remaining slices of bread on 1 side only and place them, soaked side up, on the sandwiches. Cover the skillet and cook over moderately low heat until the bread is browned on the bottom, about 3 minutes. Turn the sandwiches, adding more butter to the skillet if necssary. Cover and cook until the second side is browned and the cheese is melted, about 3 minutes longer. Transfer the sandwiches to a cutting board and letstand for 5 minutes. Cut in half and serve with the maple apples.