Heat the oven to 400°F (205°C). Lightly butter a 9 by 13 inch baking dish and set it on a foil-lined rimmed baking sheet. Chill the topping butter for 15 minutes and keep the buttermilk refrigerated. Cut a small X in the bottom of each peach, then lower them into boiling water for 30 to 45 seconds, until the skin starts to peel back. Move them to the ice water for 2 minutes, slip off the skins, and slice the fruit 1/2 inch thick.

Toss the peaches with both sugars, cornstarch, lemon juice, vanilla, cinnamon, nutmeg, and salt, and let it sit for 15 minutes. Juice should pool in the bowl, and the sugar should look dissolved.

Pour the peaches and all their juices into the dish, dot with the butter, and set the dish on the baking sheet. Bake at 400°F for 10 minutes, until the juices start to bubble at the edges.

Whisk the flour, sugar, baking powder, and salt, then cut in the cold butter until the pieces look like peas and coarse crumbs. Pour in the buttermilk and stir with a fork until a shaggy dough forms, with a few dry spots left.

Drop the dough over the hot filling in 10 to 12 large spoonfuls, leaving small gaps so steam can escape. Brush the tops with milk, then sprinkle with the coarse sugar and cinnamon.

Bake at 400°F for 30 to 35 minutes more, until the biscuit tops are deep golden, the juices bubble thickly at the edges, and a biscuit center reads 200°F. Tent loosely with foil if the tops brown before the centers cook through.

Let the cobbler rest for 20 minutes, so the juices thicken and the biscuits set. Spoon into bowls and serve warm.