Prepare the peach filling by dicing the fresh peaches and placing them in a medium saucepan over medium heat. Add granulated sugar, brown sugar, lemon juice, cinnamon, and nutmeg, cooking for about 6 minutes until the peaches soften. Mix cornstarch with water to create a slurry, stir into the peach mixture, and cook for 2-3 more minutes until thickened.

In a large mixing bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy, about 3-4 minutes. Beat in the eggs one at a time, then add the vanilla extract. Gradually mix in the flour, baking powder, cinnamon, and nutmeg until a soft dough forms. Chill in the refrigerator for at least 2 hours.

Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper. Ensure your sheets are ready as you prepare to shape the cookies.

Shape the chilled dough into balls about 1-1.5 inches in diameter. Roll each ball in a sugar and cinnamon mixture. Place the dough balls on the prepared baking sheets, spacing them about 2 inches apart. Bake for 11-13 minutes, or until the edges are set.

Allow the cookies to cool slightly, then create an indentation in each cookie and fill with the chilled peach filling. Drizzle with a glaze made from powdered sugar and a little water.