Preheat oven to 350°F (175°C). Wrap the bottom and sides of a 9-inch springform pan with heavy-duty aluminum foil to prevent water from seeping in during the water bath. Alternatively, use parchment paper on the bottom and foil around the sides.

In a medium bowl, combine the graham cracker crumbs, 1/4 cup granulated sugar, and melted butter. Mix until well combined and the crumbs are evenly moistened.

Press the mixture firmly and evenly into the bottom of the prepared springform pan. Bake for 10 minutes. Remove from oven and let cool slightly while preparing the filling. Reduce oven temperature to 325°F (160°C).

In a medium saucepan, combine the diced apples, 2 tablespoons granulated sugar, lemon juice, cinnamon, and nutmeg. Cook over medium heat, stirring occasionally, for 5-7 minutes, until apples begin to soften but still hold their shape.

In a small bowl, whisk together the cornstarch and water until smooth. Add to the apples in the saucepan and stir continuously until the mixture thickens, about 1-2 minutes. Remove from heat and let cool completely.

In a large bowl with an electric mixer, beat the softened cream cheese on medium speed until smooth and creamy, about 2-3 minutes, scraping down the sides of the bowl as needed.

Gradually add 1 cup granulated sugar, beating until just combined and no lumps remain. Do not overmix.

Beat in the eggs one at a time, mixing only until just incorporated after each addition. Scrape down the bowl after each egg.

Stir in the vanilla extract and sour cream until just combined. Be careful not to overmix the batter at this stage to prevent cracks.

Pour half of the cheesecake batter over the cooled crust in the springform pan.

Carefully spoon the cooled spiced apple mixture over the cheesecake batter, spreading it evenly but leaving a small border around the edge.

Pour the remaining cheesecake batter over the apple layer, gently spreading it to cover the apples.

Place the springform pan into a larger roasting pan. Carefully pour hot water into the roasting pan, reaching about halfway up the sides of the springform pan. This creates a water bath, essential for even baking and preventing cracks.

Bake at 325°F (160°C) for 60-70 minutes, or until the edges are set but the center still has a slight wobble when gently shaken.

Turn off the oven, crack the door open slightly, and leave the cheesecake in the oven for 1 hour to cool gradually. This helps prevent cracking.

Remove the cheesecake from the water bath and the oven. Let it cool completely on a wire rack at room temperature.

While the cheesecake is cooling, prepare the crumble topping. In a medium bowl, combine the flour, rolled oats, brown sugar, cinnamon, and pinch of salt.

Add the cold, cubed butter. Using your fingertips or a pastry blender, cut the butter into the dry ingredients until the mixture resembles coarse crumbs.

Once the cheesecake is completely cool, preheat your oven to 375°F (190°C).

Carefully sprinkle the crumble topping evenly over the cooled cheesecake.

Bake for 15-20 minutes, or until the crumble topping is golden brown and crisp.

Allow the cheesecake to cool again on a wire rack after adding the crumble. This helps it set further.

Once cooled, cover the cheesecake with plastic wrap and refrigerate for at least 6 hours, or preferably overnight, before serving. This is crucial for the cheesecake to fully set and for optimal texture.

Carefully remove the sides of the springform pan before slicing and serving. You can run a thin knife around the edge of the cheesecake before releasing the springform.