Make the filling: In a bowl, toss the berries with the sugar and cornstarch and let stand for 10 minutes.
Meanwhile, make the batter: Preheat the oven to 375°. In a medium bowl, whisk the flour with the sugar, lemon zest, baking powder and salt. In a small bowl, whisk the eggs with the milk and vanilla. Whisk the liquid into the dry ingredients until evenly moistened, then whisk in the melted butter until smooth.
Spread the filling in a 9-by-13-inch baking dish. Spoon the batter on top, leaving small gaps. Bake in the center of the oven for 1 hour, until the fruit is bubbling and a toothpick inserted into the topping comes out clean. Let cool for 1 hour before serving.