For the graham crust: Preheat the oven to 350 degrees F. Lightly spray a 9-inch springform pan with nonstick cooking spray and place on a baking sheet.
Add the graham crackers to a food processor and pulse until fine, then add the brown sugar and salt and pulse until well combined. Drizzle in the melted butter and stir until the mixture resembles damp sand. Press the crumbs into the bottom of the prepared springform pan in an even layer. Bake the crust until firm, about 10 minutes, then let cool.
For the apple crumb topping: Stir together the flour, brown sugar, cinnamon and salt in a medium bowl. Drizzle in the butter and stir until the mixture resembles wet sand, using your hands to form some clumps. Fold in the diced apple.
For the cheesecake filling: Stir together the granulated sugar, cornstarch and salt in a small bowl. Pulse together the cream cheese and sour cream in a food processor until smooth. With the processor running, add the sugar mixture 1/2 cup at a time until the mixture is smooth. Add the eggs and yolk one at a time, pulsing until well combined, then add the vanilla, scraping down the sides of the processor and giving it another pulse to make sure everything is incorporated. Pour the cheesecake filling over the top of the graham cracker crust.
Sprinkle the crumb topping over the filling. Bake the cheesecake until it is mostly set but still has a bit of a jiggle in the center, 60 to 70 minutes. Turn off the oven and leave the oven door slightly open for 1 hour. Remove the cheesecake from the oven and allow to cool to room temperature, about another hour, then refrigerate for 1 hour more. Run a paring knife around the outside before unmolding and slicing. Serve drizzled with the caramel sauce.