Add diced peaches, sugars, lemon juice, cinnamon, and nutmeg to a medium saucepan over medium heat. Cook for 6 minutes. Stir the corn starch and water together in a small bowl and stir into the peach mixture. Cook for 1 minute, then remove from the heat and stir in the 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 together, about 1 minute. Add the vanilla and eggs, and mix until just combined. Add the dry ingredients and mix until a dough forms. Chill for 30 minutes.

Scoop the dough into balls about 55 grams each 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 the butter in a mixing bowl, then stir in the remaining crumble ingredients. Crumble into small pieces on a parchment-lined pan and bake for 12-14 minutes. Set aside to cool.

Preheat the oven to 350°F again. Line cookie sheets with parchment paper. In a small bowl, stir together the rolling sugar and spices. Roll the dough balls in the cinnamon sugar mixture and place about 6 cookies on each sheet. Bake for 11-13 minutes. Let rest on the cookie sheet for 1-2 minutes and then press an indent into the centers with the back of a tablespoon. Transfer to a wire rack to cool completely.

Whisk together glaze ingredients and adjust consistency with more milk as needed.

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