Prepare oven and baking dish: Place butter in a 13- x 9–inch glass or ceramic baking dish. Place dish in oven; preheat oven to 375°F (leave dish in oven as it heats).
Make blueberry mixture: Combine blueberries, ½ cup sugar, cornstarch, and lemon juice in a large bowl, tossing well to coat.
Make cobbler topping: Whisk together flour, cornmeal, baking powder, salt, baking soda, and ¾ cup plus 2 tablespoons sugar in a medium bowl. Add buttermilk and vanilla to flour mixture; stir until well combined.
Add blueberries and topping to pan: Carefully remove baking dish from oven. Arrange blueberry mixture over butter. Pour batter over blueberries. Sprinkle remaining 2 tablespoons sugar evenly over batter.
Bake cobbler: Bake at 375°F until filling is bubbly and topping is browned, 40 to 45 minutes.
