For the crust, preheat the oven to 350º F. Butter an 8-inch square baking dish and line with parchment paper.
In a medium bowl, whisk together the flour, powdered sugar, cornstarch, zest, and salt. Cut in the butter using two forks, a pastry blender, or your hands, or pulsing in a food processor, until the mixture is crumbly. Gently bring the dough together with your hands, kneading it slightly. Put the dough into the prepared baking dish and press it into the bottom.
Bake just until starting to brown, 20 minutes. Remove the pan from the oven and let cool slightly on a wire rack while making the filling. (Keep in mind that the crust should still be warm when the filling goes on top.)
For the filling, in a medium bowl, whisk together the powdered sugar, flour, and baking powder.
Using an electric mixer, beat the eggs on high speed for 2 to 3 minutes or until their volume has tripled in size. With the mixer running on medium speed, add the sugar mixture to the eggs and mix just until blended. Add the lemon zest and juice, mixing just until incorporated. Pour the filling over the crust and bake until the filling is set in the center, 25 to 30 minutes. Cool completely on a wire rack.
Sift a little powdered sugar over the top and top with fresh berries before slicing into bars.