Preheat oven: Preheat oven to 350°F (175°C). Grease and line a 9x13-inch baking pan with parchment paper.
Prepare crust: Mix graham cracker crumbs, sugar, and melted butter until evenly coated. Press firmly into the bottom of the pan. Bake for 10 minutes, then set aside to cool slightly.
Make apple layer: Toss chopped apples with lemon juice, granulated sugar, brown sugar, apple pie spice, and salt. Spread evenly over the baked crust.
Prepare cheesecake layer: In a large bowl, beat cream cheese and powdered sugar until smooth. Add sour cream and vanilla, mixing well. Beat in eggs one at a time until just combined. Pour mixture over the apple layer.
Mix crumb topping: In another bowl, combine flour, granulated sugar, brown sugar, oats, baking powder, apple pie spice, and salt. Stir in melted butter until crumbly. Sprinkle evenly over cheesecake layer.
Bake: Bake for 50–60 minutes, or until the center is set and topping is golden brown.
Cool and chill: Let bars cool completely at room temperature, then refrigerate for at least 3 hours before slicing.
Serve: Drizzle with caramel sauce before serving, if desired.