Add diced peaches, sugars, lemon juice, cinnamon, and nutmeg to a medium sauce pan over medium heat. Cook for 6 minutes. It will be quite bubbly. 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 immediately and stir in the vanilla. Transfer to a medium 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 with the butter, about 1 minute. Add the vanilla and eggs, and mix until just combined. Add the dry ingredients last, and mix until a dough forms. Chill for 30 minutes to make it easier to roll the dough into balls.

Scoop the dough into balls 55 grams in size, with a large cookie scoop, and place in an airtight container to chill in the fridge for at least 3 hours, in the freezer for 2 hours, or in the fridge for up to 48 hours.

Preheat the oven to 350° F. Melt the butter in a medium mixing bowl. Stir in the remaining streusel ingredients. You should have a crumbly mixture. Crumble into small pieces on a parchment-lined pan and bake for 12-14 minutes, until deep golden brown. Set aside to cool. Break up the pieces further, as desired, while it cools, with a metal spatula.

Preheat your oven to 350° F. Line a few cookie sheets with parchment paper or silicone baking mats. In a small bowl, stir together the sugar and spices for coating. Roll the dough balls in the cinnamon sugar mixture. Place about 6 cookies on each sheet, with room for them to spread. Bake for 11-13 minutes. The edges should look and feel set when tapped with a fingertip, and the centers will be puffy and look just a tad underdone. Immediately place a round cutter around the outside of each cookie and swirl the cookie within it to 'scoot' them into perfect circles. This keeps them round AND helps keep them thicker. The cookies will set up as they cool. Let rest on the cookie sheet for 1-2 minutes, then press an indent into the centers with the back of a tablespoon. Let cool a few more minutes on the cookie sheet, then, once they feel sturdy enough, transfer to a wire rack to cool completely.

Whisk together glaze ingredients. Adjust glaze consistency with more milk as needed - I prefer it on the runny side here so it melts over the top of the cookie a bit.

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

Enjoy immediately once assembled! Store cookies in an airtight container in the fridge for 4-5 days. I don't recommend leaving them at room temp overnight because of the fresh fruit.