Let egg whites stand at room temperature for 30 minutes. Line a cookie sheet with parchment paper or foil; set aside. In a small bowl, stir together powdered sugar and cocoa powder; set aside. In another small bowl, stir together granulated sugar and cornstarch; set aside.
Preheat oven to 300 degrees F. In a large bowl, combine egg whites, vanilla, and cream of tartar. Beat with an electric mixer on high speed until soft peaks form (tips curl over). Gradually add the granulated sugar mixture, 1 tablespoon at a time, beating on high speed until stiff peaks form (tips stand straight). Divide mixture in half. Gradually fold cocoa powder mixture into half of the egg white mixture.
Place white and chocolate egg white mixtures into separate pastry bags, each fitted with a 1/2-inch star tip (or place in two heavy resealable plastic bags and cut a small hole in corner of each bag). Pipe twenty-four chocolate stars or mounds, each about 1 inch wide, onto prepared cookie sheet. Pipe a 1-inch-wide white star on top of each chocolate star. Repeat with remaining egg white mixtures, reversing the white and chocolate layers. Bake for 10 minutes. Turn off oven and let cool in oven for 30 minutes.