Preheat the oven to 325°F. Grease a 9-inch springform pan with butter or cooking spray.
To make the crust, combine the graham cracker crumbs, sugar, and melted butter in a bowl. Mix until the mixture is evenly moistened.
Press the mixture into the bottom of the prepared pan, using a flat-bottomed glass to press it down and create an even layer. Bake the crust for 10 minutes, then remove from the oven and let cool.
To make the filling, beat the cream cheese and sugar together in a large bowl until smooth and creamy. Add the eggs one at a time, beating well after each addition.
Mix in the vanilla extract, apple butter (or grated apples), cinnamon, and nutmeg until well combined.
Pour the filling into the cooled crust and smooth the top with a spatula.
To make the topping, melt the butter in a large skillet over medium heat. Add the sliced apples, cinnamon, and nutmeg and sauté for 5-7 minutes, until the apples are slightly softened.
Arrange the sautéed apples on top of the cheesecake filling, covering the entire surface.
Bake the cheesecake for 50-60 minutes, or until the center is just set. The cheesecake will still jiggle a bit in the center but will firm up as it cools.
Remove the cheesecake from the oven and let it cool to room temperature. Once cooled, cover the cheesecake with plastic wrap and refrigerate for at least 2 hours, or overnight, to allow it to firm up.
To serve, remove the cheesecake from the pan and slice into pieces. You can serve it with a dollop of whipped cream or a drizzle of caramel sauce, or enjoy it as is.