Butter and flour a 9-by-13-inch cake pan plus a 1-quart and 2 1/2-quart ovenproof bowl. Make both cake mixes; divide the batter among the pan and bowls. Bake at 350 degrees F until a toothpick comes out clean, about 35 minutes for the pan and small bowl, and 50 minutes for the large bowl.
Let the cakes cool 15 minutes in the pan and bowls, then unmold onto racks to cool completely. Trim the flat sides of the bowl cakes with a serrated knife to make level. Using a chef's knife, cut the 4 corners off the rectangular cake, as shown; set aside 3 of the cake triangles for the tail and beak.
Put the flat cake on a cake board or platter; use toothpicks to attach 2 cake triangles to a short end for the tail. Position the large bowl cake on the flat cake as shown, then top with the small bowl cake; insert a skewer through the cakes to secure. Trim another cake triangle to make a beak; attach to the small bowl cake with toothpicks.
Make the frosting: Heat the sugar with the cream of tartar, salt and 2/3 cup water in a saucepan, stirring, until dissolved. Beat the egg whites with a mixer until frothy. Slowly beat in the hot sugar mixture, then increase the mixer speed and beat until stiff peaks form, about 7 minutes. Beat in the vanilla and almond extracts and the food coloring.
Cover the cake with a thick layer of the yellow frosting, using the frosting to sculpt a rounded chick shape. With the edge of an offset spatula, use some frosting to extend the beak and tail.
Coat the cake with yellow sanding sugar. Refrigerate, uncovered, 15 minutes. Press the chocolate disks into the frosting for eyes. Remove the skewer and toothpicks as you cut the cake.