Preheat your oven to 350°F (175°C).
In a mixing bowl, combine 1 cup of sugar with the self-rising flour.
Stir in the milk until well combined, then mix in the melted butter.
Pour the batter into a greased 9x9-inch baking dish.
Evenly distribute the blackberries over the batter.
Sprinkle 1/4 cup of the remaining sugar over the top.
Bake for 50 minutes.
Remove the cobbler from the oven and sprinkle the remaining 2 tablespoons of sugar over the top.
Return to the oven and bake for an additional 10 minutes, or until the top is golden brown.
Serve warm, optionally topped with a scoop of vanilla ice cream.