Preheat oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish.

In a large bowl, combine the sliced peaches, 3/4 cup sugar, cornstarch, vanilla extract, and cinnamon. Toss gently until the peaches are evenly coated. Pour the peach mixture into the prepared baking dish and spread in an even layer.

To 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 with some pea-sized pieces of butter remaining. Stir in the milk until just combined; do not overmix.

Drop spoonfuls of the biscuit topping evenly over the peach mixture in the baking dish. Brush the tops of the biscuits lightly with the melted tablespoon of butter.

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

Let the cobbler cool slightly (about 15-20 minutes) before serving warm, ideally with a scoop of vanilla ice cream.