Preheat the oven to 325°F (163°C). Combine graham cracker crumbs, sugar, and melted butter. Press into a 9-inch springform pan and bake for 8–10 minutes. Let cool.

Beat softened cream cheese until smooth. Add granulated sugar and brown sugar, mix well. Beat in eggs one at a time. Add vanilla and sour cream; mix until smooth.

Pour the cheesecake filling over the cooled crust. Tap gently to remove air bubbles. Bake for 50–60 minutes, until the center is nearly set.

Turn off the oven, crack the door, and let the cheesecake cool inside for 1 hour. Remove and cool to room temperature, then refrigerate for at least 4 hours or overnight.

In a skillet, melt butter over medium heat. Add diced apples, brown sugar, cinnamon, and nutmeg. Cook for 8–10 minutes until apples soften. Let cool completely.

Before serving, top the cheesecake with the cooled apple topping. Drizzle with caramel sauce. Add crumble topping or apple slices if desired.

Slice the cheesecake with a clean knife, wiping between cuts. Serve chilled with extra caramel sauce on each slice.