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Melt half of the butter in a large frying pan and add half of the sliced apples. Sauté apples over medium heat for 4 or 5 minutes, until apples have softened slightly and are just beginning to turn golden. Arrange the sautéed apples into an even layer on top of the crust.
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Sauté the remaining apples in the remaining butter, again until the apples soften a bit. Set apples aside.
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Combine the cream cheese, eggs, sugar, and vanilla in a large mixing bowl. Beat with an electric mixer set to medium-low speed until very smooth, scraping down the sides and bottom of the bowl several times. Add the heavy cream and beat on low until thoroughly incorporated.
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Pour mixture over the apples in the prepared pan. Top the cheesecake filling with the remaining apples and press down lightly to submerge. Sprinkle the cinnamon over the top of the cheesecake.
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Fill a roasting pan** with a half-inch of water and put the pan in the oven. Set the cheesecake inside the pan and bake for 10 minutes. Reduce the oven temperature to 300 and bake an additional 65 to 75 minutes, until just the center of the cheesecake jiggles when you tap the edge of the pan.
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Remove cheesecake from the oven and cool on a wire baking rack for 2 hours, then loosely cover with plastic wrap and refrigerate overnight. Serve with whipped cream and caramel sauce if desired.