In a medium saucepan, combine diced fresh peaches, granulated sugar, brown sugar, lemon juice, cinnamon, and nutmeg. Cook over medium heat for about 6 minutes until bubbly and peaches soften. Mix cornstarch with water, stir into peach mixture, and cook for another minute until thickened. Chill in the refrigerator.

In a large mixing bowl, cream together softened butter, granulated sugar, and brown sugar until fluffy, about 3-4 minutes. Add eggs and vanilla, mixing until well combined. Gradually add flour and baking soda until smooth dough forms.

Cover cookie dough with plastic wrap and refrigerate for at least 2 hours or overnight.

Preheat oven to 350°F (175°C).

In a small bowl, mix together cold butter, flour, brown sugar, and cinnamon until coarse crumbs form. Spread on a baking sheet and bake for 10-12 minutes until golden brown. Let cool.

Shape chilled dough into balls, roll in cinnamon sugar, and place on prepared baking sheet. Bake for 11-13 minutes until edges are golden and centers slightly underdone.

Create an indent in each cookie, fill with peach filling, sprinkle crumble topping, and drizzle with glaze if desired. Let cookies cool completely before serving.