In a medium saucepan, combine diced ripe peaches, brown sugar, and cornstarch over medium heat. Stir gently and bring to a simmer for 5-7 minutes until tender.

In a mixing bowl, combine all-purpose flour, brown sugar, and ground cinnamon. Pour in melted unsalted butter and mix until a paste forms. Chill for 10 minutes.

In a large bowl, whisk together all-purpose flour, ground cinnamon, baking powder, baking soda, and salt. Set aside.

In another bowl, combine melted butter, granulated sugar, and brown sugar. Whisk until creamy, then add in the egg and vanilla extract. Mix until well-blended.

Scoop cookie dough onto a lined baking sheet, leaving space between scoops. Preheat oven to 350°F (175°C) and bake for 11-12 minutes until edges are golden.

Create a slight indentation in the center of each cookie with a spoon immediately after removing from the oven. Cool for 10 minutes on the baking sheet.

Make the frosting by beating softened cream cheese, powdered sugar, ground cinnamon, and vanilla extract together until smooth.

Once cooled, top each cookie with peach compote, a swirl of frosting, and sprinkle with crumble topping.