Preheat the oven to 375°F. Lightly grease an 8x8-inch baking dish.
In a medium bowl, combine the blackberries, blueberries, granulated sugar, cornstarch, lemon zest, vanilla extract, and lemon juice. Toss gently until evenly coated.
Transfer the berry mixture to the prepared baking dish and spread it into an even layer.
In a large mixing bowl, whisk together the flour, cornmeal, granulated sugar, baking powder, and salt.
Add the cold cubed butter and cut it into the flour mixture using a pastry cutter or your fingertips until the mixture resembles coarse crumbs.
Pour in the buttermilk and vanilla extract. Stir just until combined, being careful not to overmix.
Drop spoonfuls of the dough over the berry filling, leaving small gaps between the mounds to allow steam to escape.
If desired, brush the tops with milk or cream and sprinkle with turbinado sugar.
Bake uncovered for 35 to 40 minutes, or until the topping is golden brown and the berry filling is bubbling around the edges.
Remove from the oven and let the cobbler cool for 15 to 20 minutes to allow the filling to thicken.
Serve warm on its own or topped with vanilla ice cream.