For the Seven-Minute Frosting
3 large egg whites
1 1/4 cups sugar
5 tablespoons cold water
1/4 teaspoon cream of tartar
1 teaspoon vanilla extract
For the Cake
1 cup sifted cake flour
1 1/2 cups superfine sugar
14 large egg whites (1 3/4 cups), room temperature
1 tablespoon warm water
1/2 teaspoon salt
1 1/2 teaspoons cream of tartar
2 teaspoons vanilla extract
For the Topping
3 to 4 cups flaked coconut
Preheat the oven to 350 degrees. With a fine sieve, sift together flour and 3/4 cups sugar four times.
In the bowl of a stand mixer on medium speed, beat together egg whites and water until foamy. Add salt, cream of tartar, and vanilla; beat until soft peaks form. Increase speed to medium-high and sprinkle in remaining sugar, 1 tablespoon at a time. Beat until stiff but not dry.
Transfer to a large bowl. In six additions, sift dry ingredients over meringue, folding in quickly but gently.
Pour batter into an ungreased 10-inch tube pan with removable bottom. Smooth top with an offset spatula. Run a knife through batter to release air bubbles. Bake for 35 to 40 minutes, until golden brown and springy to touch.
Invert pan on its legs or over the neck of a glass bottle, and let cool completely, about 1 hour.
Make the frosting: In the heatproof bowl of an electric mixer set over a saucepan of simmering water, combine egg whites, sugar, water, and cream of tartar. Cook over medium heat, whisking frequently, until sugar is dissolved and mixture is foamy, about 5 minutes.
Attach the bowl to a stand mixer fitted with the whisk attachment. Beat the mixture on high speed until glossy and voluminous, about 7 minutes. Beat in the vanilla.
Carefully run a long offset spatula or knife around the inner and outer perimeter of the cake pan to release cake.
Place on a plate, bottom side up. Using a long serrated knife, carefully slice off 1 1/2 inches from the top of the cake, being sure not to break the layer. Set aside. Cut out a 1-inch-wide, 1-inch-deep c