Place 1-2/3 cup egg whites in a very large mixing bowl; let stand at room temperature for 30 minutes.
Sift powdered sugar and flour together 3 times. Beat egg whites, cream of tartar, and vanilla with an electric mixer on medium to high speed until soft peaks form (tips curl). Gradually add 1 cup 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.
Divide batter in half. Fold extract and food coloring into one half. Alternately spoon pink and white batters into an ungreased 10-inch tube pan. Gently cut through batter with a knife, swirling gently.
Bake on the lowest rack in a 350 degree F oven for 40 to 45 minutes or until top springs back when lightly touched. 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 thoroughly. Remove cake from pan.
Meanwhile, for the Pink Peppermint Frosting, place 2 egg whites in a bowl. Let stand 30 minutes. In a 2-quart saucepan mix 2/3 cup granulated sugar, corn syrup, and water. Cook and stir until mixture comes to a boil. Boil gently, without stirring, for 5 minutes. Beat egg whites using an electric mixer at medium to high speed until soft peaks form (tip curl). Slowly add hot syrup to egg whites, pouring in a steady stream, while beating. Continue beating until stiff glossy peaks form (tips stand straight). Beat in 1/4 teaspoon peppermint extract and enough red food coloring to tint frosting pink.
Frost with Pink Peppermint Frosting. Store cake leftovers in refrigerator. Makes 12 servings.