Preheat your oven to 375°F (190°C). Lightly grease an 8×8 inch baking dish.
In a bowl, gently combine the mixed berries, 1/2 cup sugar, cornstarch, lemon juice, and vanilla extract. Pour this mixture into the prepared baking dish.
To make the topping, whisk together the flour, 2 tablespoons sugar, baking powder, and salt in a medium bowl.
Cut in the cold butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs.
Pour in the milk and stir just until a shaggy dough forms. Do not overmix.
Drop spoonfuls of the biscuit dough evenly over the berry filling. Leave small gaps between the dough pieces to allow steam to escape.
If desired, brush the tops of the biscuits lightly with the beaten egg and sprinkle with coarse sugar.
Bake for 35 to 45 minutes, or until the topping is golden brown and the berry filling is bubbly.
Let the cobbler cool on a wire rack for at least 15 minutes before serving.
Notes
If using frozen berries, do not thaw them first; use them directly in the recipe. You may need to add 5-10 minutes to the baking time.
For an oat topping variation, substitute 1/4 cup of the flour with rolled oats and add 1/4 teaspoon of cinnamon to the dry topping ingredients.
Serve warm with vanilla ice cream or whipped cream.