1. Preheat your oven to 350 F (175 C). Line a 9×13-inch baking pan with parchment paper, leaving an overhang on the sides to easily lift the bars out later.

2. In a medium bowl, combine the graham cracker crumbs, 1/4 cup granulated sugar, and melted butter. Press this mixture firmly and evenly into the bottom of the prepared pan. Bake for 8-10 minutes, until lightly golden. Remove from the oven and let cool slightly.

3. While the crust cools, beat the softened cream cheese and 1/2 cup granulated sugar in a large bowl with an electric mixer until smooth and creamy. Beat in the egg and vanilla extract until just combined. Be careful not to overmix.

4. Pour the cheesecake filling evenly over the cooled graham cracker crust. Spread it gently with a spatula to ensure a smooth layer.

5. In a separate medium bowl, combine the diced apples, flour, brown sugar, cinnamon, and nutmeg. Toss gently until the apples are evenly coated. Then, use your fingertips or a pastry blender to cut in the cold butter until the mixture resembles coarse crumbs.

6. Sprinkle the apple crisp topping evenly over the cheesecake layer in the pan.

7. Place the pan in the preheated oven and bake for 40-45 minutes, or until the cheesecake is mostly set with a slight jiggle in the center and the apple topping is golden brown and bubbly.

8. Remove the pan from the oven and let the bars cool completely on a wire rack at room temperature. Once cooled, transfer the pan to the refrigerator and chill for at least 2 hours (or preferably longer) before slicing and serving.