Preheat the grill to about 375°F-400°F for indirect cooking. Place a 12" cast-iron skillet on the grill for a few minutes, just until the skillet is warm enough to melt the butter.
Add the stick of butter and brown sugar to the warm skillet. Stir until the butter is melted and the brown sugar has dissolved into the butter, creating a smooth sauce.
Arrange the peach halves in the skillet and pour the Tennessee whiskey over the top. Close the grill and let the peaches cook for about 8–10 minutes, or until they begin to soften.
Meanwhile, make the cornbread batter. In a large bowl, whisk together the cornmeal, flour, sugar, baking powder, salt, and cinnamon, if using.
In a separate bowl, whisk together the buttermilk, egg, melted butter, and vanilla. Pour the wet ingredients into the dry ingredients and stir just until combined. Be careful not to overmix.
Once the peaches have started to soften, carefully pour the cornbread batter around and partially over the peaches, leaving some of the tops of the peaches exposed.
Close the grill and cook over indirect heat at 375°F for 25–35 minutes, or until the cornbread is golden brown and a toothpick inserted into the cornbread portion comes out clean. Depending on your grill and skillet, it may need a few extra minutes.
Remove the skillet from the grill and immediately brush the top of the warm cornbread with a little melted butter.
Let the cobbler cool for about 5–10 minutes before serving. Scoop it into bowls while still warm, making sure to spoon some of the peaches and buttery Tennessee whiskey sauce into each serving. Top with vanilla ice cream and enjoy.