In a medium bowl, toss peaches with lemon juice. Add the brown sugar and cinnamon. Stir gently to combine. Let stand 30 minutes.
Drain the peaches, reserving 1 tablespoon of the juices. In a small bowl, whisk together the reserved 1 tablespoon of peach juice with the corn starch. Mix with the drained peaches.
Crumble Topping
In a medium bowl, whisk together the flour, sugar, brown sugar, cinnamon and salt.
With a pastry blender or with your fingertips, work the butter into the flour mixture until it looks like wet sand. Press the mixture together to form chunks.
Cookies
Preheat the oven to 350ºF (180ºC). Lightly spray a 12-serving muffin pan with nonstick spray.
In a medium bowl, whisk together the flour, baking powder and salt.
In a large bowl, beat the butter and powdered sugar with an electric mixer until light and fluffy.
Mix in the flour mixture on low speed or by hand, just until combined.
Press generous tablespoon size (5/8 oz or 20g each) portions of dough into the bottoms of the muffin pan cups with a pastry tamper dipped in flour. You can also use the bottom of a flat shot glass.
Top with the peaches, then the crumble topping mixture. Bake until the cookies are set and the topping is golden brown, about 15-17 minutes.
Cool the cookies completely in the pan. Once cool, remove the cookies from the pan by running a small offset spatula or paring knife around the inside perimeter of the muffin cups to loosen them. Drizzle with caramel sauce.