Preheat the oven to 375 F.
Heat a 12-inch cast iron skillet over medium heat. Melt the 2 tablespoons of butter. Add the peaches, brown sugar, granulated sugar, lemon juice, vanilla, cinnamon, nutmeg, and salt.
Cook for about 6–8 minutes, stirring occasionally, until the peaches begin to soften and release their juices.
Whisk together the cornstarch and 2 teaspoons of water. Add the cornstarch slurry to the peaches and stir for another minute until the juices slightly thicken. Remove from heat.
Make the topping by whisking together the flour, sugar, baking powder, baking soda, salt, and cinnamon in a large bowl.
Cut in the cold butter using a pastry cutter or your fingertips until the mixture resembles coarse crumbs. Mix the buttermilk and vanilla together and pour it into the flour mixture. Stir just until combined. Don’t overmix!
Use a cookie scoop to scoop the dough over the peaches and spread it out with the back of a spoon. Brush the dough with melted butter. Sprinkle the raw sugar generously over the top.
Place a baking sheet under the cast iron skillet so that it may catch any drips from the skillet. Transfer to the oven. Bake for 25-30 minutes, until top is golden brown and peaches are bubbling around the edges.
Allow the cobbler to rest 15 minutes before serving. The filling will continue to thicken as it cools. Serve warm with vanilla ice cream and salted caramel sauce.