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To Make the Crust:
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Butter the bottom and sides of a 10-inch springform pan.
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Put the gingersnaps in a food processor and whir until you have crumbs; you should have a scant 2 cups.
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Pulse in the sugar and cinnamon, if you're using it, then pour in the melted butter and pulse until the crumbs are moistened.
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Turn the crumbs into the springform pan and, using your fingertips, firmly press them evenly over the bottom and up the sides of the pan as far as they'll go.
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Put the pan in the freezer while you preheat the oven. (The crust can be covered and frozen for up to 2 months.).
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Center a rack in the oven and preheat the oven to 350 degrees F.
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Remove the pan from the freezer and wrap the bottom tightly in aluminum foil, going up the sides.
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Place the pan on a baking sheet and bake for 10 minutes, or until the crust is set and lightly browned.
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Transfer to a rack to cool while you make the apples and the filling.
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Leave the oven at 350 degrees F.
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To Make the Apples:
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Melt 2 tablespoons of the butter in a large nonstick skillet over medium-high heat.
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When the foam subsides, toss in half of the apple slices and cook, turning once, until they are golden brown, about 3 minutes.
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Sprinkle the apples with 1 tablespoon of the sugar and cook them, turning, just until coated, another minute or so.
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Scrape the apples onto a plate, wipe out the skillet and repeat with the remaining apples.
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Let the apples cool while you make the filling.
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Getting Ready to Bake:.
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Have a roasting pan large enough to hold the springform pan at hand. Put a kettle of water on to boil.
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To Make the Filling:
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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.
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Add the sugars and beat for another 2 minutes.
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Beat in the cider, vanilla, and cinnamon.
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Reduce the speed to low and beat in the eggs one by one, beating for 1 minute after each egg goes inches
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Finally, beat in the sour cream and heavy cream, beating just until the batter is smooth.
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Pour about one third of the batter into the baked crust.
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Drain the apples by lifting them off the plate with a slotted spoon or spatula, and spoon them into the pan.
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Cover with the remaining batter and, if needed, jiggle the pan to even the top.
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Place the springform pan in the roasting pan and pour in enough boiling water to come halfway up the sides of the springform pan.
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Bake the cheesecake for 1 hour and 30 to 45 minutes, covering the cake loosely with a foil tent at the 45-minute mark.
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The cake will rise evenly and crack around the edges and it should be fully set except, possibly, in the very center--if the center shimmies, that's just fine.
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Gently transfer the cake, still in the pan, to a cooling rack and let it cool to room temperature, then refrigerate it for at least 6 hours; overnight would be better.
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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.
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If desired, melt a small amount of apple jelly in the microwave and glaze the cake, or dust with confectioner's sugar.
