Place egg whites in a very large mixing bowl; let stand at room temperature for 30 minutes.
Meanwhile, sift powdered sugar and flour together 3 times. Beat egg whites, cream of tarter, and vanilla with an electric mixer on medium to high speed until soft peaks form (tips curl). Gradually add granulated sugar, about 2 tablespoons at a time, beating until stiff peaks form (tips stand straight).
Sift about one-fourth of the flour mixture over beaten egg whites; fold in gently. Repeat sifting and folding in the remaining flour mixture three more times, using one-fourth of the flour mixture each time.
Pour into an ungreased 10-inch tube pan. Gently cut through batter with a knife or a narrow metal spatula.
Bake on the lowest rack in a 350 degree F oven for 40 to 45 minutes or until top springs back when lightly touched. (The crust of the cake will be golden and slightly cracked.) Immediately invert the cake in the pan, standing the tube pan on its legs or resting the center tube over a tall-necked bottle. Cool cake thoroughly. Loosen the sides of cake from pan; remove cake from pan. To serve, use a serrated knife to slice the cake into wedges. Makes 12 servings.