Preheat oven to 325°F (163°C). Line the bottom of a springform pan with parchment paper and lightly grease the sides.

In a bowl, combine graham cracker crumbs, brown sugar, and melted butter. Stir until evenly moistened. Press firmly into the bottom of the prepared pan to form the crust. Chill while preparing the filling.

In another bowl, mix chopped apples with flour, brown sugar, cinnamon, and softened butter for the crumble topping. Use your fingers or a fork to combine until crumbly. Set aside.

In a large mixing bowl, beat softened cream cheese until completely smooth, about 2 to 3 minutes. Add brown sugar and granulated sugar. Mix until well combined and creamy.

Add sour cream, apple sauce, vanilla extract, salt, and cinnamon powder. Mix again until fully blended.

Add the eggs one at a time, mixing gently after each addition. Add the egg yolk last. Mix until just combined. Do not overmix to avoid incorporating excess air.

Pour the cheesecake batter over the prepared crust and spread evenly. Layer the chopped apples evenly on top of the batter. Sprinkle the crumble topping generously over the apples.

Place the springform pan on a baking sheet. Bake in the preheated oven for 60 to 70 minutes, or until the center is mostly set with a slight jiggle and the top is lightly golden.

Turn off the oven and crack the door open. Let the cheesecake cool inside the oven for 30 minutes. Then remove and cool at room temperature for 1 hour.

Refrigerate for at least 6 hours or overnight before serving. For clean slices, run a warm knife around the edges and wipe between cuts.

Serve chilled or with a drizzle of caramel sauce if desired.