Preheat the oven to 325 degrees F. Grease the bottom and sides of a 9-inch springform pan. In a bowl, combine the graham cracker crumbs and the sugar. Add 4 tablespoons of the butter and mix. If the crumbs seem dry, add the last tablespoon of butter. Press onto the bottom of the springform pan. Press in an even layer.
Bake for 8 to 10 minutes. The crust should be firm and golden. Remove from the oven and let cool on a wire rack. Cool the crust completely. Once cooled, wrap the outside of the pan in a heavy duty piece of aluminum foil.
Toss the apples with the lemon juice.
Melt the butter in a large skillet. Add the apples and cook for 2 minutes, over medium heat, until the apples begin to soften. Sprinkle with sugar and cook for an additional 2 to 3 minutes, until the apples soften further and begin to turn golden. Add the cream and the nutmeg and lower the heat slightly. Cook until the cream has been absorbed by the apples, about 10 minutes. Transfer the apples to a plate and let cool completely.
In the bowl of your mixer, beat the cream cheese for a minute or two until it's smooth. Add the sugar and beat at medium speed until smooth, about 2 or 3 minutes. Add the vanilla extract, the salt, the cinnamon and the ginger and beat for another minute. Add the sour cream and cornstarch and beat until blended, another minute or two. Add the eggs, one at a time, scraping the sides of the bowl after each addition.
Preheat the oven to 350 degrees F.
Layer the apples over the cooled graham cracker crust. Pour the prepared cheesecake filling over the apples. Place the pan in a large, shallow roasting pan and add hot water to come about halfway up the side of the springform pan. Carefully place the pan in the oven. Bake for between 70 and 80 minutes, until only wobbly in middle.
Remove the cheesecake from the oven and from the water bath. Peel off the aluminum foil layer and place on a rack to cool completely. Once cool, place in the refrigerator to chill overnight.
Just before serving, sift brown sugar over top of cheesecake to lightly cover the entire surface. Either using a propane kitchen torch, or by putting the cheesecake under the broiler, melt the sugar to form thin crust. (This step can be omitted and cheesecake is still excellent).