Preheat and Prepare: Preheat your oven to 375°F (190°C). Lightly butter a 9x9 inch baking dish or a similar 2-quart baking vessel.
Make the Berry Filling: In a large bowl, gently toss the mixed berries, 1/2 cup of granulated sugar, cornstarch, lemon juice, and lemon zest until the berries are evenly coated. Pour the berry mixture into the prepared baking dish and spread it into an even layer.
Prepare the Biscuit Topping: In a separate large bowl, whisk together the all-purpose flour, 1/2 cup of granulated sugar, baking powder, and salt. Add the cold, cubed butter to the flour mixture. Using a pastry blender or your fingertips, cut the butter into the flour until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.
Finish the Topping: In a small bowl, whisk together the milk and vanilla extract. Pour the wet ingredients into the dry ingredients and stir with a fork until just combined. Be careful not to overmix; a few lumps are perfectly fine.
Assemble the Cobbler: Drop large spoonfuls of the biscuit topping evenly over the berry filling. It's okay if there are some gaps; the topping will spread as it bakes. Sprinkle the turbinado sugar evenly over the dough for a crunchy, golden crust.
Bake to Perfection: Place the baking dish on a baking sheet to catch any potential drips. Bake for 35-45 minutes. You'll know the berry cobbler is done when the fruit filling is thick and bubbly and the topping is a deep golden brown. A toothpick inserted into the center of a biscuit should come out clean.
Cool and Serve: Remove the cobbler from the oven and let it cool on a wire rack for at least 15-20 minutes before serving. This allows the filling to set slightly. Serve warm, topped with a scoop of vanilla ice cream or a dollop of whipped cream.