Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish or a similar 3-quart casserole dish.

Prepare the peach filling: In a large bowl, gently toss the sliced peaches with 3/4 cup sugar, cornstarch, vanilla extract, cinnamon, and lemon juice until everything is well coated. Pour the peach mixture evenly into the prepared baking dish.

Prepare the biscuit 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 milk or buttermilk to the dry ingredients and stir just until a soft, shaggy dough forms. Avoid overmixing. Drop tablespoon-sized mounds of the biscuit dough randomly over the top of the peach filling, leaving small gaps for steam to escape.

Sprinkle the tops of the biscuits generously with extra granulated sugar for a beautiful, crunchy crust.

Bake for 40 to 45 minutes, or until the topping is golden brown and the peach filling is thick and bubbling vigorously around the edges.

Let the cobbler cool on a wire rack for at least 15 minutes before serving warm with a large scoop of vanilla ice cream.