Step 1: Preheat your oven to 350°F (175°C). Lightly grease a 9-inch springform pan. Combine graham cracker crumbs, 1/4 cup granulated sugar, and 1/2 cup melted butter in a medium bowl until well mixed. Press the mixture evenly into the bottom of the prepared springform pan. Bake for 10 minutes, then remove from oven and let cool slightly.
Step 2: Reduce oven temperature to 325°F (160°C). In a large bowl, beat the softened cream cheese and 1 cup granulated sugar with an electric mixer on medium speed until smooth. Beat in the eggs one at a time, mixing just until combined after each addition. Stir in the sour cream, vanilla extract, 1/2 teaspoon cinnamon, and nutmeg until just blended. Do not overmix.
Step 3: Pour the cheesecake batter over the cooled crust. Bake for 30-35 minutes, or until the edges are set but the center still jiggles slightly. While the cheesecake is baking, prepare the apple filling and crumble topping.
Step 4: For the apple filling, melt 2 tablespoons butter in a medium saucepan over medium heat. Add diced apples, 1/4 cup brown sugar, 1 teaspoon cinnamon, and apple pie spice. Cook, stirring occasionally, for 8-10 minutes, or until apples are tender-crisp.
Step 5: For the crumble topping, in a separate bowl, combine flour, oats, 1/4 cup brown sugar, and 1/2 teaspoon cinnamon. Cut in the cold butter pieces with your fingers or a pastry blender until the mixture resembles coarse crumbs.
Step 6: Carefully remove the cheesecake from the oven. Evenly spoon the warm apple filling over the partially baked cheesecake. Then, sprinkle the crumble topping over the apples. Return to the oven and bake for an additional 25-30 minutes, or until the crumble is golden brown and the cheesecake is mostly set.
Step 7: Turn off the oven, crack the oven door open slightly, and let the cheesecake cool inside the oven for 1 hour. This slow cooling helps prevent cracks. After 1 hour, remove the cheesecake from the oven and place it on a wire rack to cool completely at room temperature.
Step 8: Once completely cool, cover the springform pan with plastic wrap and refrigerate for at least 6 hours, or preferably overnight, before releasing from the pan and slicing. Serve chilled.