Making the Cheesecake:

1. Preheat the oven to 350 degrees.

2. On a light floured surface, roll out the pastry to fit the bottom of a 10-inch springform pan. Using the base of the springform pan as a guide, cut out a circle. Transfer the circle to a baking sheet, prick with a fork to prevent it from rising, and bake until golden, 15 to 20 minutes. Cool, and with a wide spatula, place into the springform pan. Set aside.

3. Meanwhile, caramelize the apples by cutting each quarter of apple into 5 or 6 slices, each about 1/4-inch thick. In a 10-inch skillet, melt the butter. Add the apples and 1/4 cup sugar and over medium high heat, sauté the apples until lightly caramelized, about 10 minutes stirring occasionally. The apples will be very soft. Allow to cool slightly. Spread evenly over the baked crust.

4. In an electric mixer, using the paddle, beat the cream cheese until smooth. Gradually add the remaining 3/4 cup sugar and continue to beat until the mixture is not grainy and there are no lumps. Stop the machine occasionally and scrape down the sides and under the paddle. Add the eggs, one at a time, and then the vanilla. Scrape in the chocolate that has been melted in the microwave and mix just until combined. Pour the cream cheese mixture over the apples. The filling will only reach about 1/3 up the sides of the pan. Tap the pan gently against the countertop to level.

5. Wrap heavy duty foil tightly around the outside of the pan and set the pan in the center of a large baking dish. Pour enough hot water into the larger dish to come halfway up the sides of the springform pan. Bake 55 to 60 minutes. The center will still be a little soft. That's the way you want it. Remove the foil and cool on a wire rack. Refrigerate until firm, at least 3 hours up to overnight.

6. Carefully remove the ring around the cheese cake.

Serve as is or melt 4 ounces of dark chocolate and drizzle over the top. To slice, dip a sharp knife into warm water and then cut the cake.