Preheat the oven to 350°F. Place the butter in an 8x8-inch baking dish, a 9 or 10-inch cast iron skillet, or another 2-quart baking dish. Transfer it to the oven just until melted, about 3 to 4 minutes. Keep a close eye so the butter doesn't brown too much.
In a small bowl, gently toss the blackberries with the lemon juice and the lemon zest. The lemon brightens the flavor and balances the sweetness without needing extra sugar in the fruit layer.
In a large bowl, whisk together the sugar, flour, baking powder, and salt. Stir in the buttermilk (or milk) and the vanilla just until combined. The batter will be thin.
Carefully remove the hot pan from the oven. Pour the batter directly over the melted butter. Do not stir.
Spoon the blackberries evenly over the top, along with any juices. Do not stir. Sprinkle 1 to 2 tablespoons of sugar over everything for a lightly crisp, sweet finish.
Bake for 40 to 50 minutes, or until the top is golden brown and set, and the edges are bubbling. Tent the dish loosely with foil if the top starts to get too dark before the batter is cooked through. The batter will rise up around the fruit as it bakes, creating that classic cobbler texture.
Let the cobbler rest for about 10 minutes before serving. Scoop it warm into bowls and top with vanilla ice cream.