Preheat oven to 350 degrees. Butter two 9-by-9-by-2-inch cake pans; line bottoms with parchment paper. Butter parchment and dust with flour, tapping out excess; set aside.
Whisk together flour, baking powder, and salt in a large bowl. Place shredded coconut in bowl of a food processor; pulse until finely chopped. Stir chopped coconut into flour mixture until combined; set aside.
Beat together butter and sugar in bowl of an electric mixer fitted with the paddle attachment until light and fluffy, about 4 minutes, scraping down side of bowl as needed. Add whole eggs, egg whites, and vanilla; beat until fluffy and combined. With mixer on low, add flour mixture, alternating with coconut milk, beginning and ending with flour mixture, mixing well after each addition.
Divide batter evenly between prepared pans; smooth with an offset spatula. Bake, rotating pans halfway through, until cakes are golden brown and a cake tester inserted into center comes out clean, about 45 to 50 minutes.
Transfer pans to a wire rack to cool 30 minutes. Invert cakes onto rack; peel off parchment paper. Reinvert cakes and let them cool completely, top sides up, then wrap individually in plastic wrap. Transfer to freezer for 15 to 20 minutes.
Using a serrated knife, trim tops of cake layers to make them level. Place one layer on a cake plate; spread top with 1 1/2 cups meringue. Top with remaining layer, cut side up. Using offset spatula, spread remaining frosting over entire cake, swirling to cover in a decorative fashion. Sprinkle entire cake with coconut curls, bending and curling as desired. Cake can be kept refrigerated, covered with a cake dome, up to 3 days.