In a medium saucepan, combine chopped peaches, light brown sugar, lemon juice, ground cinnamon, and nutmeg over medium heat. Cook for 5-7 minutes until the peaches soften and release their juices. Create a cornstarch slurry with cornstarch and water, add to peach mixture, and cook for an additional 2 minutes until thickened. Set aside to cool.

Whisk together all-purpose flour, granulated sugar, light brown sugar, ground cinnamon, and salt. Melt unsalted butter and pour over dry ingredients, stirring until clumpy. Spread on a baking sheet and bake at 350°F for 8-10 minutes until golden. Cool completely.

Beat softened cream cheese until fluffy. Gradually mix in powdered sugar and vanilla until smooth. Pour in heavy cream and mix until well combined. Refrigerate for at least 30 minutes.

Open the refrigerated buttermilk biscuits; cut centers to create donut shape. Heat vegetable oil in a deep skillet over medium-high heat to 350°F. Fry donuts in batches for 1-2 minutes per side until golden brown. Drain on a wire rack.

Whisk together powdered sugar, milk, and vanilla until smooth and pourable. Adjust milk for desired consistency.

Dip the tops of the warm donuts into the vanilla glaze, letting excess drip off. Set on a wire rack to let glaze set.

Spread a generous layer of cheesecake filling on each glazed donut. Spoon a helping of peach topping over, then sprinkle with cobbler crumbs. Serve immediately.