Preheat oven to 375°F. Grease a 9x13-inch baking dish with butter or nonstick spray.
In a large bowl, combine blackberries, raspberries, sugar, cornstarch, lemon juice, lemon zest, and vanilla extract. Stir gently to coat the berries without crushing them. Pour mixture into the prepared baking dish and spread evenly.
In another bowl, whisk together flour, sugar, baking powder, and salt. Add melted butter, milk, and vanilla extract. Stir until the batter is smooth and pourable, similar to pancake batter.
Pour the batter evenly over the berry mixture. It is fine if some berries peek through the top.
Bake for 40 to 45 minutes, or until the crust is golden brown and the berry filling is bubbling around the edges.
Remove from the oven and allow the cobbler to cool for at least 15 minutes before serving.
Serve warm with vanilla ice cream or whipped cream, if desired.