Preheat the oven to 375°F (190°C).
In a mixing bowl, combine the mixed berries, 1/2 cup sugar, and cornstarch. Stir until the berries are coated. Transfer the mixture to a 9-inch square baking dish.
In another mixing bowl, whisk together the flour, cocoa powder, 1/2 cup sugar, baking powder, and salt. Add in the milk, melted butter, and vanilla extract. Stir until well combined.
Drop spoonfuls of the batter over the berry mixture in the baking dish.
Pour boiling water over the top of the cobbler.
Bake in the preheated oven for 30-35 minutes, or until the cobbler is set and the top is golden brown. Serve warm.
Enjoy your Mixed Berry Chocolate Cobbler!