Prepare the baking dish and the grill: Use softened butter to thoroughly grease a large casserole or cast iron dish, all around the bottom and sides. Heat a pellet grill or smoker to 350℉ with your choice of wood or pellets.
For the blackberry filling: Add the blackberries, sugar, cornstarch, lemon juice, and vanilla extract to a large bowl. Gently fold the berry mixture until thoroughly combined. Spread filling evenly into the prepared pan. Set aside as you make the biscuit topping.
For the biscuit topping: Add the flour, sugar, baking powder, baking soda, and salt to a large bowl. Whisk to combine thoroughly.
Grate the cold butter on the largest holes of a box grater. Add the cold grated butter to the flour mixture. Use two forks or a pastry cutter to combine the butter into the flour until it resembles coarse, pea-sized crumbs (similar to the size and shape of rice cereal).
Slowly drizzle the buttermilk into the dough mixture while stirring. Mix until the dough just sticks together. Do not overmix or the biscuits will be tougher.
Assemble the cobbler: Drop dollops of the biscuit dough over the berries, leaving gaps to allow the steam to escape and the beautiful berries to show. Then brush one tablespoon of buttermilk over the biscuit topping, and sprinkle the biscuits with coarse sugar (or table sugar).
Smoke the cobbler: Cook the cobbler on the smoker for 45-55 minutes at 350°F or until the biscuit topping is golden brown and the cobbler registers at least 200°F on an instant-read thermometer. This ensures the blackberries are fully cooked, soft, and juicy, and that the juices have thickened. It also confirms the biscuit topping is cooked through and not doughy in the middle. *If the topping gets too brown, loosely tent the baking dish with foil.
Rest and serve: Remove from the grill and cool on a wire rack about 20 minutes for the juices to thicken a bit. Then serve warm with vanilla ice cream. *You can also serve the cobbler room temperature or cold if you prefer.