Bring a large pot of water to a boil. Cut a small X on the bottom of each peach. Carefully place peaches in boiling water for 30 seconds, then immediately transfer to an ice bath. The skins should easily peel off. Slice peaches into ½-inch wedges, discarding pits.
In a large bowl, gently toss sliced peaches with sugar, cornstarch, lemon juice, vanilla, cinnamon, nutmeg, and salt until evenly coated. Let the mixture sit for 15 minutes to allow the peaches to release some juice.
Preheat your oven to 375°F (190°C). Lightly butter a 9x13-inch baking dish or a 10-inch cast-iron skillet. Transfer the peach mixture to the prepared dish, spreading it evenly.
In a medium bowl, whisk together flour, sugar, baking powder, and salt. Using your fingertips or a pastry cutter, work the cold butter into the flour mixture until it resembles coarse crumbs with some pea-sized pieces remaining.
Stir in buttermilk and vanilla extract until just combined—overmixing will create a tough topping. The dough should be sticky and somewhat lumpy.
Drop spoonfuls of the biscuit mixture over the peach filling, leaving some gaps for the filling to bubble through. Sprinkle with coarse sugar if desired. Bake for 45-50 minutes until the topping is golden brown and the filling is bubbling.
Allow the cobbler to cool for at least 20 minutes before serving. This resting period allows the filling to set up properly.