Make the Crust:
Preheat the oven to 350°F (175°C). In a medium bowl, combine graham cracker crumbs, melted butter, and granulated sugar. Mix until well combined. Press the mixture into the bottom of a greased 9-inch springform pan. Bake for 8-10 minutes until set, then remove from the oven and let it cool.

Prepare the Cheesecake Filling:
In a large mixing bowl, beat the cream cheese until smooth. Add sugar and vanilla extract, and beat until creamy. Add eggs one at a time, beating well after each addition. Stir in sour cream and heavy cream until fully incorporated.

Assemble the Cheesecake:
Pour the cheesecake filling over the cooled crust. Tap the pan on the counter to remove air bubbles. Place the pan on a baking sheet to catch any drips and bake for 45-50 minutes at 325°F (160°C), or until the center is just set. Turn off the oven and let the cheesecake cool in the oven for 1 hour.

Prepare the Peach Topping:
While the cheesecake is cooling, place sliced peaches in a saucepan with brown sugar, cinnamon, and nutmeg. Cook over medium heat for 5-7 minutes until the peaches soften. Add butter and vanilla extract and cook for another 2 minutes. Mix the cornstarch and water in a small bowl, then stir into the peach mixture. Cook for another 2-3 minutes until the mixture thickens.

Assemble the Cobbler Cheesecake:
Once the cheesecake has cooled to room temperature, pour the peach topping over the cheesecake. Refrigerate the cheesecake for at least 4 hours, or overnight for best results. Serve chilled.