Cook: In a small pot, stir together diced peaches, brown sugar, and cornstarch. Cook uncovered until the peaches are tender and any excess liquid has evaporated. If the mixture looks too dry or starts burning at any point, add a few spoonfuls of water and cover with a lid for a few minutes to cook the peaches without drying them out. Transfer to a small bowl to cool completely.
Cinnamon Crumble
Prep: Preheat oven to 350°F. Line a baking sheet with parchment paper and set aside.
Mix: In a small bowl, stir together flour, brown sugar, and cinnamon. Add melted butter and stir together until it forms a thick paste. Let it sit in the fridge for 10-15 minutes to cool and firm up.
Bake: Once cooled, use your hands to break up the crumble into small chunks and spread them out across your lined baking sheet. Bake for about 5 minutes. The crumble will melt together slightly but not to worry.
Crumble up: Let the crumble cool on the pan for about 10 minutes until it crisps up. Use your hands to break the crumble apart and back into "crumbs". Let them cool completely before transferring them to a small bowl. Set aside the lined baking sheet and keep the oven on to bake the cookies later.
Cookies
Dry mixture: In a small bowl, stir together flour, cinnamon, baking powder, baking soda, and salt. Set aside.
Wet mixture: In a large mixing bowl, add the melted butter, granulated sugar, and brown sugar. Whisk together until combined. Add the egg and vanilla extract and whisk until combined and smooth.
Combine dry and wet mixture: Add the flour mixture and fold together with a spatula until just combined.
Scoop cookie dough: Use a 4-tablespoon cookie scooper to scoop the cookie dough onto your lined baking sheet, leaving about 2" in between each cookie for spreading.
Bake: Bake for 11-12 minutes or until the edges are set and the cookies are puffy. Immediately after taking them out of the oven, use the back of a spoon to press the middle of the cookies down into a large indentation. Let the cookies cool completely before adding the toppings.
Cinnamon Cream Cheese Frosting
Mix: In a small mixing bowl, add the cream cheese, powdered sugar, cinnamon, and vanilla extract. Use an electric hand mixer or a spatula to mix until it becomes smooth and creamy. Transfer the cream cheese frosting to a piping bag and cut the tip off.
Assemble the cookies
Peach topping: Add a spoonful of peach compote in the indentation of each cookie.
Top with frosting and crumble: Pipe a swirl of cream cheese frosting on top of the peach compote. Sprinkle a handful of crumble on top of each cookie to finish them off.