1. Preheat the oven to 350°F and grease a donut pan well.

2. Add the diced peaches, brown sugar, cornstarch, water, 1/2 teaspoon cinnamon, and nutmeg to a small saucepan. Cook over medium heat for 6 to 8 minutes, stirring often, until thickened. Let the peach filling cool completely.

3. In a medium bowl, beat the softened cream cheese, powdered sugar, and vanilla extract until smooth. Transfer to a piping bag or zip-top bag and chill until needed.

4. In a large bowl, whisk together the flour, 1/2 cup granulated sugar, baking powder, salt, and remaining 1/2 teaspoon cinnamon.

5. In another bowl, whisk together the milk, eggs, and 1/4 cup melted butter.

6. Pour the wet mixture into the dry ingredients and stir just until combined.

7. Spoon or pipe the batter into the prepared donut pan, filling each cavity about three-quarters full.

8. Bake for 14 to 16 minutes, or until the donuts are lightly golden and spring back when touched.

9. Let the donuts cool for 10 minutes, then remove them from the pan.

10. Mix the coating sugar and 1 teaspoon cinnamon in a shallow bowl. Brush each donut with melted butter and coat in the cinnamon sugar mixture.

11. Use a small knife or piping tip to make a hole in the side or bottom of each donut.

12. Fill each donut with cheesecake filling, then add a spoonful of cooled peach cobbler filling.

13. Serve right away or chill slightly before serving.