Preheat the oven to 325°F (163°C).
In a medium bowl, whisk together the flour and cream of tartar. Set aside. In the bowl of a stand mixer fitted with the whisk attachment, beat the egg whites until stiff but not dry, about 11/2 minutes. Use a spatula to slowly and gently fold the sugar into the egg whites. Do not beat. Add the vanilla and mix just until combined. Then add the flour mixture to the egg whites, a little at a time, stirring gently. Scrape the batter evenly into an unbuttered standard angel food cake pan.
Bake the cake for about 50 minutes, until the surface springs back when you touch it with your fingertips. Gently invert the pan on a wire rack or on top of a bottle, situating the center hole of the inverted tube pan on the neck of the bottle. Let the cake cool completely.
Slice the cooled cake horizontally into three layers. Place one cake layer, upside down, on a serving platter and spread icing over the top and sides, being especially generous as you spread the icing over the top as this will end up as filling. Sprinkle the icing on top with some of the coconut. Place the middle cake layer upside down on top of the bottom cake layer and spread icing over the top and sides, again being generous with the top layer. Sprinkle coconut over it. Place the third cake layer right side up on top of the other two stacked layers and frost the top and sides. Sprinkle the remaining coconut over the entire cake, covering it evenly. Slice and serve as soon as possible. Originally published March 30, 2012.