Preheat your oven to 325°F (163°C). Grease a 9-inch springform pan with cooking spray or butter.
In a medium bowl, combine the graham cracker crumbs, melted butter, and 1/4 cup granulated sugar. Mix until well combined. Press the mixture firmly into the bottom of the prepared springform pan to form the crust. Bake for 10 minutes, then remove from the oven and let cool.
In a large mixing bowl, beat the softened cream cheese with an electric mixer until smooth. Gradually add 1/2 cup granulated sugar and vanilla extract, mixing until well combined.
Add the eggs one at a time, mixing well after each addition. Then, mix in the sour cream until smooth.
Gently fold in the diced peaches, cinnamon, and nutmeg until evenly distributed.
Pour the cheesecake batter over the cooled crust in the springform pan.
In a small bowl, mix the brown sugar and a pinch of cinnamon. Sprinkle this mixture evenly over the top of the cheesecake batter.
Bake in the preheated oven for 50-60 minutes, or until the center is set but still slightly jiggly.
Turn off the oven and leave the cheesecake inside for an additional hour to cool gradually.
Once cooled, refrigerate the cheesecake for at least 4 hours or overnight before serving.