Preheat the oven to 350°F. In a mixing bowl, combine 1 cup sugar and the self-rising flour. Gradually stir in the milk, followed by the melted butter, mixing until the batter is smooth and cohesive.
Pour the mixture into a greased 9x9-inch baking dish, ensuring an even layer. Evenly scatter the blackberries over the top of the batter; they will settle further during baking.
Sprinkle 1/4 cup of sugar evenly across the surface. Bake the cobbler for 50 minutes.
Remove the dish from the oven and sprinkle the remaining 2 tablespoons of sugar over the partially baked crust.
Return to the oven and bake for an additional 10 minutes, or until the top develops a rich golden hue.
Serve hot, accompanied by a generous scoop of vanilla ice cream.