Heat the oven to 375°F.
Put the butter in a 9-inch square baking dish.
Slide the dish into the oven.
Let the butter melt completely while the oven finishes heating.
In a medium bowl, toss the blackberries with 1/2 cup sugar, cornstarch, and lemon juice.
In another bowl, whisk the flour, 3/4 cup sugar, baking powder, and salt.
Stir the milk and vanilla into the dry mixture until the batter looks smooth.
Carefully remove the hot dish from the oven.
Pour the batter over the melted butter.
Spoon the blackberry mixture and any juices over the batter.
Sprinkle the top with coarse sugar, if using.
Bake for 45 to 50 minutes, until the top looks golden and the fruit bubbles hard.
Let the cobbler rest for 15 minutes before serving.