For the cake: Place fresh, cold egg whites and cold water in the bowl of a heavy-duty mixer. Set aside until they are 60 degrees, or slightly below room temperature.
Pour flour, 1/2 cup granulated sugar and salt into a triple sifter or sieve. Sift onto a sheet of waxed paper; set aside. Adjust rack in lower third of oven; preheat oven to 325 degrees. Set nearby a dry, un-greased 10inch tube pan. Such a pan allows the batter to cling to the sides and rise to its full height. Place a long-necked bottle or large metal funnel nearby for inverting the baked cake.
Whip egg whites, preferably with a whisk attachment until frothy. Add cream of tartar and whip on medium speed until soft peaks form. Continue whipping and gradually add the 1 cup granulated sugar until whites have thickened and form soft droopy white peaks. Add the vanilla and lemon zest in the final moments of whipping.
Sprinkle one-quarter of the flour mixture over the whites, and with a rubber spatula, fold into the whites. Repeat this process with the remaining flour mixture, folding in only one-quarter of the flour mixture at a time. Gently pour the batter into an un-greased tube pan. Bake for 45 to 50 minutes, or until the top is light golden, the cake feels spongy and springs back when touched, and a wooden toothpick inserted in the middle comes out free of cake. Invert cake onto the neck of a bottle or funnel. Cool completely before removing from pan.
To remove cake from the pan, slip just the tip of a small metal spatula between the cake and pan. Slowly trace the perimeter to release any cake sticking to the pan. Tilt the cake pan on its side, and gently tap the bottom against the counter to loosen the cake. Rotate the cake pan, tapping as you turn it a few more times, until it appears free. Cover the cake with a rack or cardboard round, and at the same time that you invert the cake onto the work surface, tap it firmly onto the surface. Lift the pan from the cake.
For the filling: In a 1 1/2 quart heavy-bottomed saucepan, whisk briefly to combine the eggs, egg yolks, sugar, lemon juice, and lemon zest. Add the pieces of butter, and place the saucepan over medium heat, stirring constantly over the entire bottom. Cook the mixture without boiling until it begins to develop body and thicken. Remove from heat and pour through a stainless steel sieve into a bowl. Cover the surface with plastic wrap, poke a few small slits in the plastic with the tip of a paring blade allowing steam to escape while cooling. Refrigerate (curd will thicken).
For the frosting: In a 3-cup saucepan, combine the water and 2/3 cup sugar. Place over low heat to dissolve the sugar, then increase the heat to medium high and boil, without stirring, until a candy thermometer reads 235 degrees. (Wash down any sugar crystals slinging to the sides of the pan with a brush dipped in cold water.) Near the end of the boiling time, whip the egg whites with the cream of tartar until soft peaks form. Add the 2 tablespoons sugar, and continue to whip until the whites are stiff but not dry. With the mixer running, pour the syrup onto the whipped whites. Continue to whip on medium speed, about 3 minutes more. Mixture will thicken, cool and form glossy, stiff peaks. Add the vanilla. Cool to room temperature, about 8 to 10 minutes, then frost cake right away.
Assembling the dessert: Cut cake into three layers. Spread the lemon filling equally between the two layers; top with remaining layer. Frost top and sides of cake with Italian Meringue Frosting.