Peach Cobbler Cookies

Name: Peach Cobbler Cookies
Cuisine: American
Ingredients: 1 1/2 cups diced peeled and pitted fresh peaches (about 2 medium peaches); 1/3 cup firmly packed brown sugar; 1 tablespoon unsalted butter; 1/2 teaspoon vanilla extract; 1/4 teaspoon ground cinnamon; 1/8 teaspoon ground nutmeg; Pinch of salt; 2 teaspoons fresh lemon juice; 1 1/2 teaspoons cornstarch; 1 cup unsalted butter, softened; 3/4 cup firmly packed brown sugar; 3/4 cup white sugar, divided; 1 teaspoon salt; 1 teaspoon ground cinnamon; 1/4 teaspoon ground nutmeg; 1 large egg, at room temperature; 2 teaspoons vanilla extract; 1/4 teaspoon almond extract; 2 3/4 cups all-purpose flour; 1/2 teaspoon baking powder; 1/4 teaspoon baking soda