In a medium bowl, combine the sugar and lemon zest and rub together. Add the blackberries and raspberries, and toss to coat. Spoon them evenly on top of the butter in the baking dish.
MAKE THE BATTER:
In a medium bowl, whisk together the flour, sugar, baking powder and salt. Stir in the milk (the batter will be thick).
Spoon the batter on top of the berries. There might be spots it doesn't cover, and that's okay.
Bake until the fruit in the cobbler is bubbling up at the edges and a toothpick inserted into the cobbler batter on top comes out clean (about 20 minutes).
The cobbler will be too hot to eat when it comes out of the oven, so let it cool for about 30 minutes. Serve topped with vanilla ice cream or whipped cream.