Preheat your oven to 350°F (175°C) and line a 9x13 inch baking pan with parchment paper, leaving an overhang on the sides for easy removal.

Combine graham cracker crumbs, ¼ cup granulated sugar, and melted butter in a bowl, then press this mixture firmly into the bottom of the prepared pan to form the crust; bake for 10 minutes and let cool slightly.

Beat softened cream cheese and ¾ cup granulated sugar with an electric mixer until smooth and creamy, then gradually beat in the eggs one at a time.

Mix in the sour cream and vanilla extract until just combined, ensuring not to overmix, then pour the cheesecake filling over the cooled crust and spread evenly.

In a separate bowl, toss diced apples with brown sugar, cinnamon, and lemon juice, then gently spoon the apple mixture evenly over the cheesecake filling.

Bake for 35-40 minutes, or until the edges are set and the center jiggles only slightly, then turn off the oven and leave the bars inside with the door ajar for 1 hour to cool slowly.

Remove the pan from the oven, let it cool completely on a wire rack at room temperature, then refrigerate for at least 4 hours, or preferably overnight, before slicing and drizzling with caramel sauce.