Position an oven rack in the center of the oven. Heat the oven to 350F degrees. Line an 8-inch square baking pan with foil, allowing the foil to extend over the rim by 2 inches. Spray the foil with non-stick spray.
To make the crust:
In a small bowl, whisk together 2/3 cup of the flour, cornmeal, baking powder and salt until well blended; then set it aside.
In another small bowl, beat together the eggs and egg white.
Take 1 tablespoon of the egg mixture and place it in a cup, along with the teaspoon of water, and stir it together. Set aside both egg mixtures.
With an electric mixer on low speed, beat the butter in a medium bowl until creamy. Add 1/4 cup of the granulated sugar and beat until blended. Beat in the egg-water mixture. Add the flour mixture and beat until blended. (The dough will be crumbly.)
Press the dough mixture evenly onto the bottom and 1/4-inch up the sides of the foil-lined pan. Bake until golden brown, 20 to 25 minutes.
Make the filling while the crust bakes:
In a medium bowl, whisk together the remaining 1/3 cup flour and 3/4 cup granulated sugar. Whisk in the remaining egg mixture until blended. Add the lemon zest and lemon juice and whisk until well blended.
Remove the crust from the oven and place on a rack. Reduce the oven temperature to 300F degrees. Pour the lemon mixture over the hot crust. Bake until the filling is set, 20 to 25 minutes.
Remove from the oven and place on a wire rack to cool. Refrigerate until cold, at least 1 hour and up to 8 hours.
When ready to serve, sprinkle the chilled bars with confectioners' sugar. Use the foil to lift the slab of bars from the pan. Cut into 16 even squares.