In a medium saucepan, combine diced peaches, granulated sugar, brown sugar, lemon juice, cinnamon, and nutmeg. Cook over medium heat for about 6 minutes until bubbly and fragrant. Mix cornstarch with water and stir into the mixture, cooking for an additional minute until thickened. Cool in the fridge.

In a large mixing bowl, cream together unsalted butter, granulated sugar, and brown sugar until light and fluffy. Add eggs and vanilla extract, mixing well. Gradually incorporate flour, baking powder, cinnamon, and nutmeg, stirring until just combined. Chill the dough for at least 2 hours or overnight.

In a separate bowl, mix melted butter, brown sugar, and flour until crumbly. Spread on a baking sheet and bake at 350°F for 12-14 minutes until golden brown. Cool completely before breaking into crumbs.

Preheat your oven to 350°F. Roll chilled dough into 1.5-inch balls, roll in granulated sugar, and place on a parchment-lined baking sheet. Bake for 11-13 minutes, until edges are set but centers are underbaked. Use a cookie cutter to create indents.

After cooling slightly, spoon peach filling into the indents and sprinkle crumbled topping over. Whisk together milk and powdered sugar, drizzle over cookies, and let set before serving.