Preheat the oven to 350°F. Butter the bases of three 8-inch round cake pans, line each with parchment paper, then butter the parchment.
In a mixing bowl, lightly whisk the egg whites. Incorporate the milk and vanilla extract, blending well; set aside.
In the bowl of a stand mixer, combine the cake flour, granulated sugar, baking powder, and salt. Beat on low speed to break up any clumps. Gradually add the unsalted butter and coconut milk. Continue mixing on low until ingredients are just combined, then increase to medium speed and beat until the batter is fluffy and light, approximately 2 minutes.
Gradually pour in the egg white mixture in two to three additions, scraping down the bowl after each. Evenly distribute the batter among the prepared pans.
Bake for 30 minutes or until a tester comes out clean. Let the cakes rest in their pans for 10 minutes before transferring to a wire rack to cool completely.
To assemble, place one cake layer (flat side up) on an 8-inch cake board. Spread with 1 cup of frosting, ensuring even coverage to the edges. Sprinkle with ½ cup of flaked coconut. Repeat layering with the second cake and frosting. Position the final layer on top and frost both the top and sides of the cake.
Spread the remaining 1½ cups of coconut on a baking tray. Support the cake in one hand over the tray, using the other hand to press the coconut onto the sides. Once fully coated, transfer the cake to a serving plate and sprinkle any remaining coconut over the top. Chill for a minimum of one hour to set the frosting.