Prepare the crust: Preheat your oven to 325°F (163°C). In a medium bowl, combine the graham cracker crumbs, sugar, and melted butter. Stir until the mixture resembles wet sand. Press it into the bottom of a 9-inch springform pan, creating an even layer. Bake for 8-10 minutes, then remove from the oven and set aside to cool.

1. Make the cheesecake filling: In a large bowl, beat the softened cream cheese with an electric mixer until smooth and creamy. Add the sugar and vanilla extract and beat until combined. Add the eggs one at a time, beating well after each addition. Stir in the sour cream until fully incorporated.
2. Bake the cheesecake: Pour the cheesecake filling into the cooled crust. Tap the pan gently on the counter to remove any air bubbles. Bake for 50-60 minutes, or until the center is set but still slightly jiggly. Turn off the oven and let the cheesecake cool in the oven with the door slightly open for 1 hour. Afterward, refrigerate the cheesecake for at least 4 hours or overnight for the best texture.
3. Prepare the caramel apples: In a large skillet, melt the butter over medium heat. Add the apple slices and cook for 5-7 minutes, until they begin to soften. Stir in the brown sugar, cinnamon, nutmeg, and salt. Continue to cook for another 5 minutes, or until the apples are tender and caramelized. Remove from heat and set aside to cool slightly.
4. Make the caramel sauce: In a medium saucepan, melt the butter over medium heat. Add the brown sugar and heavy cream, and stir until the sugar dissolves. Bring to a simmer and cook for 3-5 minutes until thickened. Remove from heat and stir in the vanilla extract and salt.
5. Assemble the cheesecake: Once the cheesecake has cooled and set, top it with the caramelized apples. Drizzle with the homemade caramel sauce just before serving.