Line a 9x13 inch baking pan with two sheets of aluminum foil, going in opposite directions so the foil hangs over all four sides. This foil sling is what lets you lift the whole batch out cleanly later. Grease the foil lightly with cooking spray. Preheat the oven to 325 degrees F.

Make the crust. Combine the graham cracker crumbs, 1/3 cup sugar, and melted butter in a bowl and stir until every crumb is evenly coated and the mixture looks like wet sand. Pour it into the prepared pan and press it into an even, firm layer using the bottom of a flat glass or measuring cup. Bake the crust alone for 10 minutes, then set it aside while you prepare the filling and compote.

Make the apple compote. Melt 1 tablespoon of butter in a medium saucepan over medium heat. Add the diced apple, brown sugar, cinnamon, ginger, and cloves. Stir and cook for 5 to 7 minutes until the apple is tender but still holds its shape. Stir the cornstarch into the cold water and pour it into the pan. Cook for 1 minute more until the liquid thickens and coats the apple. Remove from heat and let it cool while you make the filling.

Beat the room temperature cream cheese with a hand mixer or in a stand mixer on medium speed for 2 to 3 minutes until completely smooth with no lumps at all. Scrape the bowl and beat again if you see any. Add the 3/4 cup sugar and beat on medium until fully incorporated, about 1 minute. Add the vanilla and sour cream and mix until smooth.

Add the eggs one at a time on low speed, mixing just until each one disappears into the batter before adding the next. Do not beat on high or whip air into the batter after the eggs go in. Overmixing at this stage can cause the cheesecake to crack as it bakes. Scrape the bowl once more and give it a final gentle stir by hand.

Pour the cream cheese filling over the prebaked graham crust and spread it into an even layer with a rubber spatula. Spoon the cooled apple compote over the top in an even layer, distributing the apple pieces across the whole surface. Do not press the apple into the filling, just let it sit on top.

Bake at 325 degrees F for 33 to 38 minutes. The edges should look set and slightly puffed, and the center should barely jiggle when you give the pan a gentle shake. It will look slightly underdone in the very center, and that is correct. It firms up as it cools.

Remove the pan from the oven and let it cool on a wire rack for 1 hour at room temperature. Then transfer the whole pan to the refrigerator and chill for at least 3 hours, or overnight. Do not try to slice it early. Once fully chilled, grab the foil overhang and lift the whole slab out of the pan onto a cutting board. Slice into bars with a sharp knife, wiping the blade clean between cuts for neat edges.