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

Scoop the dough into balls (approximately 55 grams each) and place in an airtight container to 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 medium mixing bowl. Stir in the remaining streusel ingredients. Crumble into small pieces on a parchment-lined pan and bake for 12-14 minutes, until deep golden brown. Break up the pieces further while cooling.

Preheat the oven to 350°F. Line cookie sheets with parchment paper. 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. Let cool slightly on the cookie sheet, then press an indent into the centers with the back of a tablespoon.

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