Preheat the oven to 375°F (190°C). Lightly grease a 9 x 13-inch (23 x 33 cm) baking dish. Set the dish on a rimmed baking sheet if it is shallow or likely to bubble over.

Peel, pit, and slice the peaches approximately 1/2 inch (1.25 cm) thick. Keep the slices relatively even so they soften at the same rate.

Place the peaches in a large bowl: Add the granulated sugar, cornstarch, lemon juice, vanilla, cinnamon, and salt. Fold gently until the peaches are evenly coated and no dry patches of cornstarch remain.

Transfer the peach mixture and all accumulated juices to the prepared baking dish. Spread the peaches evenly and bake uncovered for 10 minutes. The fruit should begin releasing juice around the edges but will not yet be fully tender.

While the peaches prebake, whisk together the flour, granulated sugar, baking powder, and salt in a medium bowl.

Add the cold butter cubes: Use a pastry cutter or two forks to work the butter into the flour until the mixture contains pea-size pieces with some smaller crumbs. Avoid blending until completely smooth, as visible butter pieces help create a tender topping.

Pour in the cold buttermilk and vanilla: Stir gently with a flexible spatula just until a shaggy dough forms. The dough should be soft and moist with no large pockets of dry flour. Do not knead or overmix it.

Carefully remove the baking dish from the oven: Drop the biscuit dough over the hot fruit in 10 to 12 portions, leaving small gaps between them so steam can escape and the fruit remains visible.

Sprinkle the optional coarse sugar evenly over the biscuit topping. Keep most of the sugar on the dough rather than allowing it to collect in the fruit juices.

Return the dish to the oven and bake for 30 to 35 minutes. The filling should bubble around the edges and through the gaps, and the biscuit topping should be golden brown.

Insert a toothpick into the thickest part of a biscuit, avoiding the peach filling. It should come out clean or with a few moist crumbs. Wet dough indicates that the cobbler needs another 3 to 5 minutes.

If the biscuits become deeply browned before their centers are done, place a loose sheet of aluminum foil over the dish and continue baking.

Transfer the cobbler to a wire rack and let it cool for at least 20 minutes. The filling will be extremely hot and initially loose but will thicken as it stands.

Spoon the warm cobbler into shallow bowls, making sure each serving includes peaches, sauce, and biscuit topping.