Add diced peaches, sugars, lemon juice, cinnamon, and nutmeg to a medium saucepan over medium heat. Cook for 6 minutes, stirring occasionally. Stir the cornstarch and water together in a small bowl and stir into the peach mixture. Cook for 1 minute more, then remove from heat and stir in vanilla. Transfer to a bowl to cool completely in the fridge.

In a large bowl, cream the softened butter with an electric mixer on medium-high speed. Add the sugars and mix until creamed, about 1 minute. Add vanilla and eggs, mix until just combined. Add dry ingredients and mix until dough forms. Chill for 30 minutes.

Scoop dough into balls of 55 grams and place in an airtight container. Chill in the fridge for at least 3 hours or in the freezer for 2 hours.

Preheat the oven to 350°F. Melt butter in a bowl, stir in remaining crumble ingredients until crumbly. Crumble onto a parchment-lined pan and bake for 12-14 minutes until golden brown.

Preheat oven to 350°F. Line cookie sheets with parchment paper. Stir together sugar and spices for coating. Roll dough balls in the mixture. Bake for 11-13 minutes until edges are set but centers are puffy. Optional: Use a round cutter to shape cookies while hot.

Whisk together glaze ingredients. Adjust consistency with more milk as needed for a runny glaze.

Spoon peach cobbler filling into cooled cookies, sprinkle on crumble, and drizzle with glaze.

Enjoy immediately after assembling! Store cookies in an airtight container in the fridge for 4-5 days.