In a medium saucepan over medium heat, combine peeled and diced fresh peaches, granulated sugar, brown sugar, cinnamon, and nutmeg. Cook for about 6 minutes until bubbly and softened. Mix cornstarch with water to create a slurry, then stir into peach mixture to thicken. Remove from heat to cool.

In a large bowl, cream together softened butter, granulated sugar, and brown sugar until light and fluffy. Add eggs one at a time, followed by vanilla extract. Gradually mix in flour, baking powder, baking soda, cinnamon, and nutmeg to form a thick dough. Chill dough for at least 2 hours.

Preheat oven to 350°F (175°C) and line baking sheet with parchment paper. Scoop chilled dough into tablespoon-sized balls, roll in a sugar-cinnamon mix, and place on baking sheet with spacing. Bake for 11-13 minutes until edges set and centers remain soft. Cool on wire rack.

Once cookies are completely cooled, create an indent in the center of each. Spoon peach filling into each indentation and drizzle glaze over cookies.