Step 1: Preheat your oven to 350°F (175°C). In a medium bowl, combine the graham cracker crumbs, 1/4 cup granulated sugar, and melted unsalted butter. Mix until the crumbs are uniformly moistened.
Step 2: Press the mixture firmly and evenly into the bottom and up the sides of a 9-inch springform pan. Bake for 8-10 minutes, or until lightly golden. Remove from the oven and let cool slightly.
Step 3: Reduce oven temperature to 325°F (160°C). In a large mixing bowl, beat the softened cream cheese on medium speed until smooth and creamy, about 2-3 minutes. Scrape down the sides of the bowl.
Step 4: Add 1 cup granulated sugar, flour, and vanilla extract. Beat on low speed until just combined. Add the eggs one at a time, beating on low speed after each addition just until incorporated.
Step 5: Pour the cheesecake filling into the cooled graham cracker crust. Gently tap the pan on the counter a few times to release any air bubbles.
Step 6: Bake for 45-55 minutes, or until the edges are set but the center still has a slight jiggle. Turn off the oven, prop the door open slightly, and let the cheesecake cool in the oven for 1 hour.
Step 7: Remove from the oven, let cool completely on a wire rack, then cover and refrigerate for at least 4 hours, or preferably overnight, until firm.
Step 8: For the apples: In a large skillet over medium heat, melt 2 tablespoons unsalted butter. Add the sliced apples and cook for 5-7 minutes, until they start to soften.
Step 9: Sprinkle in the brown sugar, cinnamon, nutmeg, and a pinch of salt. Continue to cook for another 5-10 minutes, or until the apples are tender-crisp and the sauce has thickened slightly. Remove from heat and let cool completely.
Step 10: For the caramel: In a heavy-bottomed saucepan, combine 1 cup granulated sugar and water over medium heat. Stir until sugar dissolves, then stop stirring. Cook until the syrup turns a deep amber color (8-12 minutes).
Step 11: Carefully remove from heat. Slowly and cautiously pour in the warm heavy cream while whisking constantly. Add the butter and vanilla extract, whisking until smooth. Let cool completely; it will thicken.
Step 12: Once the cheesecake is chilled and firm, gently spread the cooled cinnamon apples evenly over the top. Drizzle generously with the homemade caramel sauce. Slice and serve.