Preheat your oven to 350°F (175°C).
In a large mixing bowl, combine 1 cup of sugar with the self-rising flour.
Gradually stir in the milk, followed by the melted butter, until the mixture is smooth and well-blended.
Pour this batter into a greased 9 x 9-inch baking dish, spreading it evenly.
Scatter the blackberries over the top. As the cobbler bakes, the berries will sink further into the batter.
Evenly sprinkle 1/4 cup of sugar over the top, reserving the additional 2 tablespoons for later use.
Bake in the preheated oven for 50 minutes.
Remove from the oven and sprinkle the remaining 2 tablespoons of sugar over the cobbler.
Return to the oven and bake for another 10 minutes, or until the top is golden and crisp.
Serve warm, topped with a scoop of vanilla ice cream if desired.