Preheat your oven to 375°F (190°C). Lightly grease a 9-inch pie dish or an 8x8 inch baking dish.

Prepare the filling: In a large bowl, gently combine the blackberries, 3/4 cup sugar, cornstarch, lemon juice, and vanilla extract until the berries are evenly coated. Pour the mixture into the prepared baking dish.

Prepare the topping: In a separate medium bowl, whisk together the flour, 1/4 cup sugar, baking powder, and salt. Cut in the cold butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs.

Add the 1/2 cup milk to the dry ingredients and stir just until a shaggy dough forms. Do not overmix. Drop spoonfuls of the batter randomly over the top of the blackberry filling, leaving some berries showing through.

Brush the tops of the biscuit mounds lightly with the reserved tablespoon of milk or cream.

Bake for 35 to 40 minutes, or until the topping is golden brown and the filling is bubbly and thickened. Let cool for at least 15 minutes before serving.