Preheat your oven to 350°F (175°C). Place the sugar into a food processor or blender and pulse for about 30 seconds to create a finer texture. Measure out 1 cup of the processed sugar and sift it together with the cake flour using a fine-mesh sieve. Repeat the sifting process two more times to ensure a light, airy mixture. Set aside.

In the bowl of a stand mixer or a large mixing bowl, combine the egg whites and warm water. Using the whisk attachment on low speed, mix until the egg whites become foamy. Add the salt, cream of tartar, and vanilla extract. Increase the mixer speed to medium and continue beating until soft peaks begin to form. Gradually increase the speed to medium-high, and slowly sprinkle in the remaining ½ cup of sugar, one tablespoon at a time. Continue beating until stiff but glossy peaks form. The peaks should stand upright without collapsing and maintain a shiny appearance.

Remove the bowl from the mixer. Using a spatula, gently fold the sifted flour and sugar mixture into the beaten egg whites in six separate additions. Be sure to fold carefully and efficiently to avoid deflating the batter.

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

Invert the cake pan immediately—either using the pan's built-in legs or by placing it upside down over the neck of a glass bottle—and allow the cake to cool completely for about 1 hour. Once cool, carefully run a thin, sharp knife around the outer edges of the cake and along the center tube to loosen it. Invert the cake onto a serving plate, bottom side up, and set aside while you prepare the frosting.

Place a heatproof mixing bowl (preferably the bowl of your stand mixer) over a saucepan filled with simmering water. The water should be hot with small bubbles forming, but not boiling.

Add the egg whites, sugar, water, and cream of tartar to the bowl. Whisk the mixture constantly over medium heat until the sugar has fully dissolved and the mixture is foamy—this should take about 5 minutes.

Attach the bowl to the stand mixer fitted with the whisk attachment. Beat on high speed until the frosting becomes thick, glossy, and voluminous. This process may take up to 7 minutes. Once the frosting reaches the desired texture, whisk in the vanilla extract.

Use a long serrated knife to carefully slice off the top 1½ inches of the cooled cake, being careful not to break it. Set this top layer aside. Cut a channel into the bottom layer of the cake, approximately 1 inch wide and 1 inch deep, positioned halfway between the center and outer edge. Fill this channel with a portion of the frosting, then spread more frosting across the exposed surface of the bottom layer. Sprinkle sweetened flake coconut over the frosted layer.

Gently replace the reserved top layer of cake. Spread the remaining frosting evenly over the top and sides of the cake.

Generously sprinkle the sweetened flake coconut over the entire cake, gently pressing it into the frosting along the sides and across the top to ensure full coverage.