Step 1: Prepare the Crust

1. Preheat Oven: Preheat your oven to 325°F (163°C).
2. Mix Ingredients: In a medium bowl, combine the graham cracker crumbs, sugar, and melted butter. Stir until the mixture resembles wet sand.
3. Form the Crust: Press the mixture into the bottom of a 9-inch springform pan, pressing firmly with the back of a spoon to form an even layer.
4. Bake the Crust: Bake for 10 minutes, then set aside to cool while you prepare the filling.

Step 2: Prepare the Cheesecake Filling

1. Beat Cream Cheese: In a large mixing bowl, beat the softened cream cheese on medium speed until smooth and creamy, about 2-3 minutes.
2. Add Sugar and Sour Cream: Add the sugar and sour cream, beating until well combined and smooth.
3. Add Eggs: Add the eggs one at a time, mixing on low speed after each addition just until combined. Avoid overmixing, as this can cause cracks in the cheesecake.
4. Add Vanilla and Flour: Mix in the vanilla extract and flour, blending until just combined.

Step 3: Prepare the Apple Filling

1. Cook the Apples: In a medium skillet, melt 1 tablespoon of butter over medium heat. Add the apple slices, brown sugar, cinnamon, and nutmeg.
2. Sauté: Cook the apples for 5-7 minutes, stirring occasionally, until they are tender and caramelized. Set aside to cool slightly before adding to the cheesecake.

Step 4: Assemble the Cheesecake

1. Pour Filling over Crust: Pour the cheesecake filling over the prepared crust, spreading it evenly with a spatula.
2. Add Apple Layer: Arrange the caramelized apples on top of the cheesecake filling in an even layer. Gently press them down to partially submerge them in the filling.

Step 5: Make the Crumble Topping

1. Combine Ingredients: In a small bowl, mix the flour, rolled oats, brown sugar, and cinnamon. Pour in the melted butter and stir until the mixture forms a crumbly texture.
2. Sprinkle on Top: Sprinkle the crumble topping evenly over the apple layer.

Step 6: Bake the Cheesecake

1. Bake: Place the cheesecake in the oven and bake for 60-70 minutes, or until the edges are set and the center is slightly jiggly.
2. Cool Gradually: Turn off the oven and crack the door open slightly. Allow the cheesecake to cool in the oven for 1 hour to prevent cracking.
3. Chill: Remove the cheesecake from the oven and cool completely on a wire rack. Once cool, refrigerate for at least 4 hours or overnight.

Step 7: Serve and Garnish

1. Slice and Serve: Remove the cheesecake from the springform pan and slice into pieces.
2. Garnish: Drizzle with caramel sauce and add a dollop of whipped cream, if desired, before serving.