Preheat oven to 350 degrees.

Add the sugar to a food processor or blender and pulse for about 30 seconds. Add 1 cup of the sugar and the sifted cake flour to a fine-mesh wire sieve, and sift together. Repeat 2 more times. Set aside.

Add the egg whites and warm water to the bowl of a stand mixer or large mixing bowl. Using a whisk attachment, on low speed, whisk together the egg whites and water until foamy. Add the salt, cream of tartar, and vanilla. Beat on medium speed until soft peaks form. Increase the speed to medium-high and sprinkle in the remaining ½ cup of sugar, 1 tablespoon at a time. Keep whisking until the egg whites are stiff but not dry.

Remove the bowl from the mixer and add the sifted flour and sugar, by sprinkling the mixture over the egg whites in six additions, using a spatula to gently but quickly fold in.

Pour the batter into an ungreased 10-inch tube pan with a removable bottom. Smooth the top with an offset spatula. Run a knife through the batter to release any air bubbles. Bake for 35 to 40 minutes, until the top of the cake is golden brown and springs back when touched.

Invert the cake pan on its legs, or if it doesn’t have legs invert it onto the neck of a glass bottle, and allow it to cool completely, about 1 hour. Carefully run a thin sharp knife around the outside edges of the cake and the inside tube to loosen it from the sides of the pan and invert it, bottom side up, onto a cake plate. Set aside and make the frosting.

Place a heatproof bowl from a stand mixer over a saucepan of simmering water. (The water should have air bubbles around the edges but not be boiling).

In the bowl, combine the egg whites, sugar, water, and cream of tartar. Cook over medium heat, whisking frequently until the sugar is dissolved and the 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 it becomes glossy and voluminous. This could take up to 7 minutes. Whisk in the vanilla.

Using a long serrated knife, carefully slice off the top 1 ½ inches of the cake, making sure to not break it, and set it aside. Cut a channel, 1-inch wide and 1-inch deep halfway between the center and the edge of the cake. Fill the channel with the frosting and spread the frosting over the entire layer and sprinkle the coconut over the layer.

Gently place the reserved cake layer on top of the bottom cake layer. Spread the remaining frosting over the sides and top of the cake.

Sprinkle the coconut over the top and sides of the cake, gently patting the coconut into the frosting on the sides.