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 1/2 cup melted butter. Mix until the crumbs are evenly moistened.
Step 2: Press the mixture firmly into the bottom and up the sides of a 9-inch pie plate. Bake for 8-10 minutes, or until lightly golden. Remove from oven and let cool completely on a wire rack.
Step 3: In a large bowl, using an electric mixer, beat the softened cream cheese on medium speed until smooth and creamy. Scrape down the sides of the bowl.
Step 4: Gradually add 1/2 cup granulated sugar, beating until just combined. Beat in the egg and vanilla extract until just incorporated.
Step 5: Pour the cheesecake mixture into the cooled graham cracker crust. Spread evenly. Bake for 20-25 minutes, or until the edges are set but the center still jiggles slightly. Remove from oven, let cool completely on a wire rack, then refrigerate for at least 1 hour.
Step 6: While the cheesecake chills, peel, core, and slice the apples into 1/4-inch thick slices. In a large skillet over medium heat, melt 2 tablespoons unsalted butter.
Step 7: Add the sliced apples, brown sugar, cinnamon, nutmeg, and a pinch of salt to the skillet. Cook, stirring occasionally, for 8-12 minutes, or until the apples are tender-crisp. Remove from heat and let cool completely.
Step 8: Once the cheesecake layer is firm and the apples are cool, spread the spiced apple mixture evenly over the cheesecake layer in the pie crust.
Step 9: Drizzle generously with 3/4 cup caramel sauce. If desired, sprinkle with chopped toasted nuts.
Step 10: Cover the pie loosely with plastic wrap and refrigerate for at least 4 hours, or preferably overnight, to allow all the layers to meld together perfectly.