Preheat the oven to 350°F (175°C). Use a 9-inch springform pan at least 3 inches deep. Reserve a little of the crust's melted butter to grease the sides; mix the remaining crust butter with the fine cookie crumbs. Press into the bottom of the pan and bake for 3–5 minutes. Let cool while preparing the filling.

Beat the softened cream cheese until smooth, then mix in the sugar, vanilla, and sour cream. Add the eggs one at a time, mixing on low just until incorporated. If desired, tint about ½ cup of batter with green food coloring. Grease the pan sides with the reserved butter, wrap the outside with several layers of heavy-duty foil, and pour the batter over the crust. Swirl in the tinted batter.

Set the springform pan in a larger roasting pan. Carefully add hot water to come about halfway up the outside of the springform pan, below the foil's top edge. Bake for 10 minutes at 350°F, then reduce the oven to 275°F (135°C) and continue baking for about 1 hour 50 minutes. Begin checking after 60 minutes at the lower temperature; ovens and pans vary. The edges should be set and the center should wobble gently as one piece, not ripple like liquid. Check the center with an instant-read thermometer; it should reach 160°F (71°C). Once done, turn off the oven, crack the door, and let the cheesecake cool for 1 hour. Remove it from the water bath, then refrigerate for at least 4 hours or overnight.

While the cheesecake bakes or cools, make the fried apples. Melt the butter in a skillet over medium heat. Add the apples, both sugars, cinnamon, nutmeg, and lemon juice. Cook, stirring, for about 8–10 minutes, until the apples are tender. Mix the cornstarch with 1 tablespoon cold water, stir it into the bubbling apples, and cook for 1–2 minutes until thickened. Transfer to a shallow container, cool briefly, then refrigerate within 2 hours. Let the topping cool completely before assembling.

After the cheesecake is out of the oven, heat the oven to 350°F (175°C). Mix the crumb topping's melted butter, brown sugar, flour, and cinnamon until evenly combined. Break into small crumbs and spread in a thin, even layer on a parchment-lined baking sheet. Bake for 15–20 minutes, stirring and breaking up large clumps halfway through, until the topping is browned and baked through. Do not taste the unbaked mixture. Cool completely, then break into crumbs; it will firm as it cools. This topping must be baked before it goes on the chilled cheesecake.

To make the caramel, heat the sugar in a saucepan over medium heat until melted and amber-colored. Carefully stir in the butter, then slowly add the cream while stirring; the mixture will bubble vigorously. Simmer for 1–2 minutes until smooth. Let cool until pourable but no longer hot.

Once the cheesecake has chilled and set, release it from the springform pan. Top with the cooled apples, fully baked and cooled crumbs, and cooled caramel. Refrigerate for another 30 minutes before slicing.