Mix 1 ½ cups graham cracker crumbs and 2 tablespoons sugar in a bowl.
Add 6 tablespoons unsalted butter, melted, and stir until the mixture resembles wet sand.
Press firmly into the bottom and halfway up the sides of a 9-inch springform pan (23cm). Bake at 350°F (180°C) for 8–10 minutes, until it starts to dry and color. Leave the oven on.
Caramelize the apples:
Peel, core and slice 2 ½ pounds tart apples. Toss in a bowl with 1 tablespoon lemon juice.
Melt 3 tablespoons unsalted butter in a large skillet over high heat. When it begins to foam, add the apple slices and cook for 3 or 4 minutes, until they begin to soften. This time will depend on the thickness of the slices.
Sprinkle ¼ tablespoons white sugar, ⅓ cup light brown sugar, reduce the heat to medium and continue to cook for about 3 minutes, until the sugar dissolves and a syrup is formed. Don't move them around much. Some pieces will be darker than others.
Add 2 tablespoons water and 1 tablespoon rum or Calvados (or more water), being very careful if using alcohol not to let any drop fall into the fire, or it will ignite. Let alcohol evaporate and cook a bit longer, but don’t let the apples soften too much or start to turn into mush, or the liquid evaporate completely. You want syrup to serve.
Sprinkle ¾ teaspoon ground cinnamon, lightly stir and transfer to a colander over a shallow bowl and drain the syrup. Let both preparations cool while preparing the cheesecake batter.
Prepare the cheesecake layer:
Beat 24 ounces regular cream cheese, softened, in a large bowl until smooth.
Add ¾ cup sugar and mix well with a whisk or a handheld electric mixer. The mixture should be completely smooth.
Add 3 eggs, one at a time, mixing with a whisk just to get an even, well-integrated batter. Don’t beat.
Add 1 tablespoon lemon juice, 1 teaspoon vanilla and ¼ cup regular sour cream and mix until fully incorporated. Don’t beat.
Distribute ¾ of the apples on top of the baked cheesecake base, reserving the rest for the topping. Carefully pour the cheesecake mixture over the apples, covering completely.
Bake at 350ºF (180°C) for 15 minutes.
Without opening the oven for long, reduce the temperature to 300°F (150°C) and continue baking for 45 minutes, until the edges are set and the center still jiggles slightly when the pan is gently shaken.
Turn off the oven and leave the cheesecake inside, without opening the door, for 1 hour.
Remove from the oven, place on a wire rack and carefully run a smooth-bladed knife around the edges so there's less chance of cracks. Cool completely. Cover in plastic wrap and refrigerate for at least 8 hours or overnight before serving. Store covered in the refrigerator for up to 5 days.
Make the whipped cream topping (optional):
Beat ¾ cup heavy cream with 2 teaspoons powdered sugar in a medium bowl until firm peaks form. Place in a piping bag and decorate the cold cheesecake with whipped cream rosettes around the edges. You can fill the bag and refrigerate it before piping, so you have a firmer topping.
Serve with the rest of the apples on top and pour syrup over each slice. A drizzle of caramel sauce is also a great addition.
Notes
If you want more apple syrup, I recommend you make an extra half-filling recipe separately from the original one. You will have more apples after you drain the syrup, which you can use to top pancakes, waffles or oatmeal. Cool down gradually. Cheesecakes don't like sudden temperature changes, and this is key for creaminess and to avoid cracks. Let the baked cheesecake sit inside the turned-off oven as instructed, then cool completely at room temperature before covering the pan with plastic wrap and chilling for several hours in the fridge. This ensures the best texture.Always remove the cheesecake cold, straight from the fridge. If softened, it will be too complicated to transfer it to the serving plate, and it will probably crack, or worse, break.