Add diced peaches, granulated sugar, brown sugar, lemon juice, cinnamon, and nutmeg to a medium saucepan over medium heat.

Cook for about 6 minutes until bubbly and softened.

Mix cornstarch and water in a small bowl, then stir into the peach mixture.

Cook for 1 minute until slightly thickened.

Remove from heat and stir in vanilla extract.

Transfer to a bowl and chill in the refrigerator until completely cooled.

In a large mixing bowl, beat softened butter until creamy.

Add granulated sugar and brown sugar and mix for about 1 minute until fluffy.

Mix in vanilla extract and eggs until combined.

Add flour, baking soda, baking powder, salt, cinnamon, and nutmeg.

Mix until a dough forms.

Chill dough for 30 minutes.

Scoop dough into balls about 55 grams each.

Place in an airtight container and chill in the refrigerator for at least 3 hours or up to 48 hours.

Preheat oven to 350°F (175°C).

Melt butter in a bowl and stir in brown sugar, flour, and cinnamon.

Spread crumble onto a parchment-lined baking sheet.

Bake for 12–14 minutes until golden brown.

Let cool and break into small pieces.

Preheat oven to 350°F and line baking sheets with parchment paper.

Mix sugar, cinnamon, and nutmeg in a small bowl.

Roll dough balls in the cinnamon sugar mixture.

Place about six cookies per sheet.

Bake for 11–13 minutes until edges are set and centers are slightly soft.

Use a round cutter to gently shape the cookies while warm.

Press an indent in the center of each cookie with the back of a spoon.

Whisk powdered sugar, vanilla extract, cinnamon, and milk until smooth.

Adjust thickness with additional milk if needed.

Spoon peach filling into the center of each cooled cookie.

Sprinkle crumble topping over the peaches.

Drizzle with glaze if desired.