Coat a 1 1/2-quart and a 2 1/2-quart ovenproof bowl with cooking spray. Prepare the cake mixes; divide the batter between the bowls. Bake at 325 degrees F, about 45 minutes for the small bowl and about 55 minutes for the large bowl. Let cool, then remove from the bowls.
Make the frosting: Heat the sugar with the cream of tartar, salt and 1/3 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 beat until stiff peaks form, about 7 minutes.
Remove about 2 cups of the frosting to a small bowl and fold in enough yellow food coloring to make a bright egg yolk color (about 8 to 10 drops); set aside.
Set the cakes rounded-side down. Trim the tops with a serrated knife so the cakes sit at the same height.
Flip the cakes over and slice off a 3-inch piece from each one as shown; reserve the trimmings.
Line up the cut sides of the cakes, then trim the sides of the smaller cake as shown to make an oval shape. Flip the cakes over.
Use a little white frosting to stick the cakes together, then cover completely with a thin layer of frosting; freeze 20 minutes. Cover with the remaining white frosting, making it as smooth as possible.
Crumble the reserved cake trimmings and mix with just enough of the prepared yellow frosting to moisten. Clump the mixture together and mound on the wider end of the cake to form the yolk.
Put the remaining yellow frosting in a pastry bag fitted with a large star tip. Pipe in a spiral over the cake mound. Sprinkle with the crushed strawberry "paprika" and top with the sour straw "chives."