Chop the peaches, rinse the blueberries and blackberries under cold water. Preheat oven to 220 degrees with a rack in the center. Whisk the brown sugar and cornstarch together in a large bowl.
Add berries and chopped peaches, stir.
Place the berry mixture in a shallow baking dish and bake for about 12 minutes, until the mixture is warm.
Meanwhile, stir together the flour, baking powder and salt in a large bowl. Add chilled butter, chopped into chunks, and stir with your fingertips or a dough cutter until the mixture looks like large crumbs. Add milk and stir until a smooth dough is obtained.
Remove the dish from the oven and place 12 lumps of dough on the hot berry mixture. Stir in the remaining 1 teaspoon of brown sugar and granulated sugar and sprinkle on top of the dough.
Bake the cobbler until golden brown, 25 to 30 minutes.