Preheat oven to 425 degrees F (220 degrees C). Lightly oil an 8-inch square or 8-inch round baking dish.
Combine blackberries and 1 cup sugar in a saucepan. Whisk warm water and cornstarch together in a bowl until smooth; stir into blackberries. Bring blackberry mixture to a boil, reduce heat to medium, and cook, stirring frequently, until berries break down and thicken, 5 to 10 minutes.
Whisk flour, 1/3 cup sugar, baking powder, and salt together in a bowl. Cut butter into flour mixture using a pastry cutter or your hands until mixture resembles coarse crumbs. Stir milk into flour-butter mixture until a smooth, thick batter forms.
Pour berry mixture into the prepared baking dish. Drop spoonfuls of batter over the berries.
Bake in the preheated oven until a firm and golden crust forms, 30 to 40 minutes.