Peach Filling and Topping:
Combine sliced peaches and lemon juice in a pot. Add brown sugar, sugar, corn starch, cinnamon, and nutmeg. Stir and cook over medium-high heat for 8-10 minutes until peaches soften. Set aside to cool.

Graham Cracker Crust:
Prepare a 9-inch springform pan by lining the bottom with parchment paper and lightly spraying with non-stick spray. Wrap the outside of the pan with heavy-duty aluminum foil to prevent water leakage. Set aside. Preheat oven to 425F (220C).
Mix graham cracker crumbs and melted butter until evenly moistened. Press this mixture into the bottom of the prepared springform pan. Place the pan in the refrigerator or freezer while preparing the filling.

Cheesecake Filling:
Beat softened cream cheese with sugar and vanilla until smooth. Mix in sour cream and corn starch. Add slightly beaten eggs and mix on low speed just until combined; avoid overmixing.

Assemble and Bake Part 1:
Pour about half of the cheesecake mixture over the crust and smooth the top. Place the springform pan in a larger roasting pan. Carefully pour hot water into the roasting pan until it reaches halfway up the sides of the springform pan. Bake for 13-15 minutes, until the top is set.
Reduce oven temperature to 350F (175C).
Remove the cheesecake from the oven. Arrange a thin layer of the cooled peach filling over the cheesecake layer. Reserve any leftover peaches and their sauce for serving.
Spoon the remaining cheesecake mixture over the peaches and smooth the top. Return the pan to the oven. Bake for about 40-45 minutes, or until the center of the cheesecake is just slightly jiggly.

Cobbler Topping:
While the cheesecake bakes, prepare the cobbler topping. Whisk together flour, brown sugar, sugar, baking powder, and salt. Cut in cold butter using a fork until the mixture resembles coarse crumbs. Add boiling water and stir until just combined. Set aside.

Assemble and Bake Part 2:
When the cheesecake is nearly done (the top is set), carefully remove it from the oven. Arrange the remaining peach filling on top of the cheesecake. Dollop spoonfuls of the cobbler topping mixture over the peaches. Place the cheesecake back in the oven for 20-25 minutes, until the cobbler topping is golden brown.
Turn off the oven. Leave the cheesecake in the oven with the door slightly ajar for one hour. Remove from the water bath and place on a wire rack to cool completely to room temperature. Chill in the refrigerator for 4-6 hours or overnight.

Finishing and Serving:
Once chilled, carefully run a thin knife around the edge of the cheesecake to loosen it from the springform pan. Remove the ring and transfer the cheesecake to a serving plate.
Before serving, top the cheesecake with the reserved peaches and their sauce. You may gently reheat the sauce if desired before drizzling it over the cake. Store any leftovers in the refrigerator.