Begin by preheating your oven to 350°F (175°C). Crush your cookies in a food processor until fine, then combine them with the melted butter. Press the mixture into the bottom of a springform pan to form the crust and bake for about 5 minutes. Remove from the oven and let it cool while you prepare the cheesecake filling.

In a large mixing bowl, beat the softened cream cheese until smooth. Add the sugar and continue mixing until creamy, then blend in the vanilla extract and sour cream. Add the eggs one at a time, mixing until fully incorporated. For a touch of color, set aside a small portion of the batter and tint it with a few drops of green food coloring. Brush the inside of your springform pan lightly with melted butter, wrap the outside with foil, pour in the batter, and swirl the green-tinted mixture on top for a marbled look.

Prepare a water bath by placing the springform pan inside a larger baking tray and adding enough hot water to cover the bottom. Bake the cheesecake for 10 minutes at 350°F, then reduce the oven temperature to 275°F and continue baking for about 50–60 minutes. When done, let the cheesecake cool in the oven with the door slightly open, then chill it in the refrigerator for at least 4 hours or overnight.

While the cheesecake cools, make the fried apples. In a skillet over medium heat, melt the butter, then add the chopped apples, both sugars, cinnamon, nutmeg, and lemon juice. Stir frequently until the apples are tender and coated in a syrupy glaze. Sprinkle in the cornstarch and cook another minute until thickened, then remove from heat and set aside.

Next, prepare the crumb topping by melting the butter and stirring in the brown sugar, flour, and cinnamon. Mix by hand until crumbly, like the topping on a coffee cake. To make the caramel sauce, heat the sugar in a saucepan over medium heat, whisking constantly until it melts and turns a deep amber color. Add the butter, stir until incorporated, then slowly pour in the heavy cream while whisking. Simmer for a minute or two until the sauce thickens, then let it cool slightly.

Once the cheesecake has chilled and set, remove it from the springform pan. Spoon the fried apples evenly over the top, sprinkle on the crumb topping, and drizzle generously with the caramel sauce. Chill for another 30 minutes before slicing.