Preheat the oven to 350°F (175°C). Lightly grease a 9x13-inch baking dish or a deep 8x8-inch dish, depending on how thick you want the cobbler.
Prepare the blackberry mixture: In a medium bowl, combine the blackberries, ¾ cup sugar, lemon juice, and cornstarch (if using). Stir gently and let sit while you prepare the batter.
Make the batter: In a separate bowl, whisk together the flour, ½ cup of the sugar, baking powder, and salt. Stir in the milk, melted butter, and vanilla until smooth. The batter will be thināthis is normal.
Assemble the cobbler: Pour the batter into the prepared baking dish. Do not stir. Gently spoon the blackberry mixture evenly over the batter. Again, do not mix them togetherāthe batter will rise around the berries as it bakes. Sprinkle the remaining ½ cup sugar evenly over the top.
Bake for 40ā45 minutes or until the top is golden brown and the berry juices are bubbling. If the top browns too quickly, tent loosely with foil during the last 10 minutes.
Cool slightly before serving. The cobbler will set as it cools.
Serve warm, ideally with a scoop of vanilla ice cream or a dollop of whipped cream.