Place egg whites in a medium mixing bowl; let stand at room temperature for 30 minutes.
Line two large cookie sheets with brown paper or foil; set aside.
Add vanilla and cream of tartar to egg whites. Beat with an electric mixer on medium to high speed until soft peaks form (tips curl). Gradually add sugar, 2 tablespoons at a time, beating until stiff, glossy peaks form (tips stand straight) and sugar dissolves. Quickly beat in peppermint extract. Tint pink with several drops of red food coloring, if desired.
Using a pastry tube with a large star decorating tip, pipe cookies onto prepared cookie sheets (form cookies about 1-1/2 inches in diameter). Bake in a 300 degree F oven for 15 minutes. Turn off oven and let cookies dry in oven with door closed about 30 minutes. Remove from cookie sheets. Cover and store in a dry place. Makes 45 cookies.