Heat oven; sift cake flour and sugar: Preheat the oven to 350°F. With a fine sieve, sift together flour and ¾ cups sugar four times.

Beat egg whites: 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.

Sift dry ingredients over meringue: Transfer to a large bowl. In six additions, sift dry ingredients over meringue, folding in quickly but gently.

Transfer batter to tube pan: 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 over bottle to cool: Invert pan on its legs or over the neck of a glass bottle, and let cool completely, about 1 hour.

Make 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.

Beat frosting and add vanilla: 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.

Release cake from pan; slice cake horizontally and cut channel in bottom half: 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 ½ 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 channel halfway between the center and edge of cake.

Spread icing over bottom layer: Spread icing into channel and over entire layer. Sprinkle with coconut.

Add top layer and spread with frosting: Gently place reserved layer on top and ice with remaining frosting. Sprinkle with coconut and serve immediately.