Preheat the oven to 375°F and spray 2 round cake pans with cooking spray. Cut two rounds of parchment paper and line the cake pans; set aside.
Whisk together 3 cups cake flour, 1 tablespoon baking powder, and salt in a bowl and set aside.
Cream together 1 cup butter and 1 2/3 cups sugar until light and creamy-looking.
Add the dry ingredients to the butter mixture, alternating with 1 cup coconut milk.
Whip 8 large egg whites and 1/3 cup sugar.
Carefully fold the whipped egg whites into the cake batter.
Divide the batter between the two prepared pans. Bake until lightly golden brown and the cakes spring back when touched and pull away from the sides, 30 to 35 minutes.
Place the cakes on a cooling rack and cool for 10 minutes. Flip the cakes out of the pans onto the rack, discard the parchment, and let cool completely.
Start the frosting: Combine 1 1/2 cup sugar, 3 egg whites, 1/3 cup coconut water, and cream of tartar in a heatproof bowl over a pot of simmering water. Mix with a hand mixer until combined and then whip until glossy. Remove from heat, add coconut extract and beat. Let cool.
Spread 1 cup of frosting on one layer, then place the other layer on top of it and spread the remaining frosting on the cake's top and sides.
Sprinkle 2 cups shredded coconut over the cake, then gently press into the sides of the cake.
Refrigerate the cake for 30 minutes before serving,