Position a rack in middle of oven and preheat to 400°F. Whisk sugar and cornstarch in a large bowl. Add berries and lemon juice to bowl and toss to coat. Transfer berry mixture to a 2-quart baking dish about 2" deep. Set aside to macerate while making dough.

Whisk 1 1/3 cups flour, 1/4 cup sugar, baking powder, and salt in a medium bowl. Add butter; using fingertips, rub in butter until coarse meal forms. Add lemon zest.

Add 1/2 cup milk and stir just until dough forms. Gather dough into a ball and transfer to a lightly floured surface. Gently knead 5 or 6 times, then pat or roll dough into a 9x6" rectangle. Cut dough in half lengthwise, then crosswise 3 times, forming 8 rectangles. Arrange biscuits 1/2" apart over fruit. Brush tops lightly with milk and sprinkle with sugar.

Bake until fruit is bubbling and biscuits are just cooked through and golden all over, about 35 minutes. Let cool at least 30 minutes; serve with vanilla ice cream. DO AHEAD: Cobbler can be made up to 3 hours ahead. Rewarm at 350° if desired.