Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish or a deep 9-inch square dish.

Prepare the peach filling: In a large bowl, toss the peach slices with 3/4 cup sugar, lemon juice, cinnamon, nutmeg, and cornstarch until the peaches are evenly coated. Pour the 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.

Combine wet ingredients: In a small cup, mix together the milk and vanilla extract. Pour this mixture into the dry ingredients and stir just until a shaggy biscuit dough forms. Do not overmix.

Assemble and bake: Drop spoonfuls of the biscuit dough randomly over the top of the peach filling, leaving small gaps between the mounds to allow steam to escape. Sprinkle the top evenly with extra granulated sugar for a crisp finish.

Bake for 40 to 45 minutes, or until the topping is golden brown and the peach filling is bubbling thickly around the edges. Let cool for at least 15 minutes before serving warm.