Preheat oven to 350 degrees. Butter a 9-cup oval gratin dish. Next in a small bowl, mix together sugar, cinnamon, and nutmeg and set aside. Peel and core the apples and cut into thin slices and toss with lemon. Place apple slices, water and brown sugar into a small saucepot and cook until the apples are tender and the sugar has all dissolved. Slice the bread into triangles and layer into the buttered baking dish. Brush the bread with melted butter and liberally sprinkle the cinnamon/sugar mixture over the bread. Top with 1/2 of the cheddar cheese and 1/2 of the apples. Repeat with second layer of bread and apples. Whisk together 4 large eggs, 2 3/4 cups milk, and 1/4 teaspoon salt and pour over the pudding. Let the pudding sit, refrigerated, for 1 hour. Bake for 30 minutes and then sprinkle with the remaining 1/2 cup cheddar. Let bake until cheese is melted and golden, about 15 minutes.