Instructions

1. Prepare the crust:  Preheat oven to 350° F. Butter the sides and bottom of a 9-inch springform pan. You can slip a 9-inch round of parchment paper on the bottom and butter that too if you have one handy - it will help to keep the crust from sticking when serving. Wrap the bottom of the pan with 2 layers of aluminum foil or 1 layer of wide heavy-duty aluminum foil. The foil should reach ¾ up the height of the pan.
2. In a medium bowl, stir the graham cracker crumbs, sugar, and salt together. Pour the melted butter over the crumbs and stir with a fork until all of the dry ingredients are uniformly moist. Pour the moistened crumbs into the springform pan and use your fingers or the bottom of a glass or measuring cup to pat an even layer of crumbs over the bottom of the pan and just about ½-inch up over the lip of the bottom of the pan where it meets the side. (At this point, the crust can be covered and frozen for up to 2 months.)
3. Place the pan on a baking sheet. Bake for 10 minutes. Allow the crust to cool completely on a wire rack while you make the cheesecake. Reduce the oven temperature to 325° F.
4. To make the apples: Melt 2 tablespoons of the butter in a large nonstick skillet over medium-high heat. When the foam subsides, toss in half of the apple slices and cook, turning once, until they are golden brown, about 3 minutes. Sprinkle the apples with 1 tablespoon of the sugar and cook them, turning, just until coated, another minute or so. Scrape the apples onto a plate, wipe out the skillet and repeat with the remaining apples. Let the apples cool while you make the filling.
5. Getting ready to bake: Have a roasting pan large enough to hold the springform pan at hand. Put a kettle of water on to boil.
6. To make the filling: Working with a stand mixer, preferably fitted with a paddle attachment, or with a hand mixer in a large bowl, beat the cream cheese on medium speed, scraping down the bowl often, for about 4 minutes, or until it is velvety smooth. Add the sugars and beat for another 2 minutes. Beat in the cider, vanilla, and cinnamon. Reduce the speed to low and beat in the eggs one by one, beating for 1 minute after each egg goes in. Finally, beat in the sour cream and heavy cream, beating just until the batter is smooth.
7. Pour about one third of the batter into the baked crust. Drain the apples by lifting them off the plate with a slotted spoon or spatula, and spoon them into the pan. Drizzle half of the salted caramel sauce over the apples. Cover with the remaining batter and, if needed, jiggle the pan to even the top.
8. Dollop the remaining salted caramel sauce by the teaspoon in a random pattern into the top of the batter and then with a sharp knife, swirl the caramel around from dot to dot and in between.
9. Place the springform pan in the roasting pan and pour in enough boiling water to come halfway up the sides of the springform pan.
10. Bake the cheesecake for 1 hour and 30 minutes, covering the cake loosely with a foil tent at the 45-minute mark. The cake will rise evenly and may crack around the edges, and it should be fully set except, possibly, in the very center. If the center shimmies, that's just fine.
11. Turn off the oven and leave the cheesecake inside. Stick the handle of a wooden spoon between the door and the oven to vent the oven. This will allow the cheesecake to set up a little as it's starting to cool down.
12. Gently transfer the cake, still in the pan, to a cooling rack and let it cool to room temperature, then remove the foil (be careful as there may be some residual water inside from the water bath), and refrigerate it for at least 6 hours; overnight would be better.
13. Run a blunt knife around the edges of the pan to loosen the crust, open the pan's latch and release and remove the sides. Slice into 12 to 16 slices and serve with extra salted caramel sauce drizzled over the top.