Preheat oven to 350°F.
Butter bread slices thinly on one side, using about half the stick of butter.
In a small bowl, stir together 1/4 cup of the granulated sugar, along with the cinnamon and nutmeg. Sprinkle evenly over buttered sides of the bread.
Arrange bread on baking sheets, buttered side up, and toast in batches on the middle rack of the oven until just golden, or about 10-15 minutes. Cool cinnamon toast on racks. Cross-cut toast diagonally to create four triangles from each slice. Set aside 24 of the toast triangles for later.
Peel, quarter and core apples. Cut apples lengthwise into thin slices. In a medium bowl, toss the apple slices with lemon juice.
In a large heavy skillet over medium-high heat, melt the remaining half stick of butter. As the butter is melting, add the brown sugar and stir. Add apple slices and water. Cook mixture over moderate heat, covered, for about 10 minutes, stirring occasionally. Remove lid and continue to cook apples until just tender and most of the liquid is evaporated, or about an additional 5 minutes.
Using the reserved 2 tablespoons of butter, grease a large baking dish (about 3-quart size). Arrange one layer of cinnamon toasts in the dish, breaking triangles as necessary to fit. Spoon half of the apple mixture evenly over toasts. Sprinkle everything evenly with 1/2 cup shredded Cooper Cheese. Top with another layer of cinnamon toasts and remaining apple mixture. Arrange the 24 reserved cinnamon toast triangles, sugared sides up, overlapping slightly in a ring around the edge of the dish.
In a medium bowl, whisk together eggs, remaining 1/2 cup granulated sugar, milk and salt. Pour evenly over apples. Cover the bread pudding and chill for at least 1 hour, up to 1 day.
When bread pudding has chilled, preheat oven to 350°F.
Uncover the bread pudding and bake for 30 minutes.
Sprinkle remaining 1/2 cup Cooper Sharp cheese over the top and bake another 15 minutes until cheese is melted and bread pudding is baked through.
Make the caramel sauce:
In a small sauce pan over low heat, melt 1 stick unsalted butter. Add brown sugar and whipping cream, stirring until the sugar is dissolved, approximately 5 minutes. Remove from heat and let cool.
Serve bread pudding warm with a dollop of vanilla ice cream and a drizzle of caramel sauce.