1. 1. Prepare the Crust: Preheat your oven to 350 F (175 C). In a medium bowl, combine the graham cracker crumbs and 1/4 cup granulated sugar. Pour in the melted butter and mix until the crumbs are evenly moistened. Press this mixture firmly into the bottom of an 8×8-inch baking pan lined with parchment paper, leaving an overhang on two sides to help lift the bars out later. Bake for 8-10 minutes, then remove from the oven and let cool slightly while you prepare the filling.

2. 2. Make the Cheesecake Filling: In a large mixing bowl, beat the softened cream cheese with 1/2 cup granulated sugar until smooth and creamy, with no lumps. Beat in the eggs one at a time, mixing just until combined after each addition. Stir in the vanilla extract. Be careful not to overmix the batter once the eggs are added, as this can introduce too much air and cause cracks in the cheesecake.

3. 3. Assemble the Bars: Pour the cream cheese mixture evenly over the partially baked graham cracker crust. Return the pan to the oven and bake for 25-30 minutes, or until the edges are set but the center still has a slight jiggle.

4. 4. Prepare the Apple Topping: While the cheesecake is baking, melt 2 tablespoons of butter in a medium skillet over medium heat. Add the finely diced apples, brown sugar, cinnamon, and nutmeg. Cook for 5-7 minutes, stirring occasionally, until the apples are tender but still hold their shape. In a small bowl, whisk together the flour and water to create a slurry. Stir this slurry into the apple mixture and cook for another 1-2 minutes, until the sauce thickens slightly. Remove from heat.

5. 5. Add the Apple Topping: Once the cheesecake is out of the oven, let it cool in the pan on a wire rack for about 10-15 minutes. Then, carefully spoon the warm apple topping evenly over the cheesecake layer.

6. 6. Chill and Serve: Allow the bars to cool completely at room temperature, then transfer the pan to the refrigerator and chill for at least 2 hours, or preferably 4 hours to overnight, until fully set. Once chilled, use the parchment paper overhangs to lift the entire cheesecake from the pan. Cut into 12-16 squares and serve.