For the cake: Preheat the oven to 350 degrees F. Lightly spray a 9-by-13-inch baking dish with nonstick spray.
Place the egg whites in the bowl of an electric mixer fitted with the whisk attachment. Whip on medium-high speed until stiff peaks form, 4 to 5 minutes. Gently remove the egg whites from the mixer to a medium bowl and set aside.
Combine the sugar and butter in the bowl of the mixer and cream together, scraping down the sides of the bowl occasionally, until light and fluffy, about 3 minutes. Add the vanilla and whip an additional 30 seconds to incorporate.
In another medium bowl, sift together the cake flour, baking powder and fine salt. With the mixer on low speed, pour half the coconut milk into the butter and sugar mixture. Turn the mixer off and add half the dry ingredients. Mix on low speed until the dry ingredients are just incorporated, scraping down the sides of the bowl as needed. Repeat this process with the remaining coconut milk and dry ingredients.
Using a rubber spatula, gently fold the whipped egg whites into the batter. Pour the batter into the greased baking dish. Bake until golden brown and a toothpick inserted in the center comes out clean, about 45 minutes. Allow the cake to cool completely.
For the coconut syrup: Place the coconut water and honey in a microwave-safe container. Microwave until the honey is dissolved, about 1 minute. Poke the surface of the cake with the tines of a fork. Brush the cake with the warm syrup. Sprinkle the flaked coconut evenly over the cake.
For the frosting: Place the sugar and coconut water in a small saucepan. Bring to a boil over medium-high heat. Let boil for 2 minutes before you add it to the egg whites (see below).
Meanwhile, place the egg whites in the bowl of an electric mixer fitted with the whisk attachment. Whip on medium-high speed until soft peaks form, 2 to 3 minutes. Add the vanilla extract. Reduce the speed to the lowest setting and slowly pour the hot sugar syrup into the egg whites. Continue to drizzle until all the syrup has been added. Then, turn the mixer up to high speed and whip until the frosting becomes thick, glossy and holds stiff peaks, about 3 minutes more. The frosting should be used to decorate the cake immediately.
Scrape the frosting onto the center of the cake in a large mound. Push outward with an offset spatula, spreading the frosting over the surface of the cake and making cloudlike swirls to cover completely.
The cake can be decorated with piping gel or melted chocolate as desired! Make the frosting just before it is time to decorate the cake because it dries slightly as it sits.