Preheat oven to 325 degrees F. Prepare a 9-inch round springform pan by spraying it with nonstick spray and lining the bottom with a round piece of parchment paper.
Make the crust: In a medium bowl, combine graham cracker crumbs, brown sugar, and melted butter. Stir together until crumbs are evenly moistened. Add mixture to the bottom of the springform pan, using the bottom of a cup or jar to spread the crumbs into an even layer, covering both the bottom and some of the sides.
Bake crust for 10 minutes, then allow to cool while you make the rest of the cheesecake.
Make the cheesecake: Add cream cheese and brown sugar to the bowl of your stand mixer fitted with the paddle attachment. Beat until fully combined, light, and creamy, about 1 to 2 minutes.
Add in eggs, sour cream, vanilla, and salt. Beat again until combined, about another 1 to 2 minutes.
Pour half the cheesecake batter over the cooled graham cracker crust. Carefully spread apple butter in a thin, even layer over top of the cheesecake batter. (*SEE NOTES*) Pour remaining half of cheesecake filling over the apple butter. Make the apple crumb topping: To a medium bowl, add cold butter, flour, oats, brown sugar, cinnamon, and salt. Using a pastry cutter or fork, cut the butter into the other ingredients until it creates a pea-sized crumb.
Top cheesecake batter with an even layer of diced apples. Then, sprinkle the apple crumb topping on top of the apples.
Bake cheesecake for 1 hour 15 minutes, or until golden brown on top and just slightly jiggly in the middle. If you have an instant read thermometer, the internal temperature should register 150 to 155 degrees F.
Remove cheesecake from the oven and cool completely on a wire rack as close to the oven as possible. The residual heat will help it cool gradually, so it does not crack. Once cheesecake has cooled, cover it and refrigerate for at least 4 hours, but preferably overnight.
Serve sliced cheesecake with a scoop of vanilla ice cream and a drizzle of caramel sauce over top.