Make the cake: Heat oven to 325°. Grease three 9" cake pans with butter and line with parchment paper; set aside. Whisk flour, desiccated coconut, baking powder, and salt in a bowl; set aside. Place sugar and egg yolks in the bowl of a stand mixer fitted with a paddle attachment; mix until fluffy. Add mayonnaise, coconut milk, buttermilk, coconut and vanilla extracts; mix until combined. With the motor running, slowly add dry ingredients until batter forms; transfer batter to a bowl and set aside. Clean stand mixer bowl and add egg whites; replace paddle with whisk attachment. Whip whites until soft peaks form; fold into cake batter. Divide batter between prepared pans, smoothing tops with a rubber spatula; drop pans lightly on a counter to expel large air bubbles. Bake until a toothpick inserted in middle comes out clean, about 25 minutes. Let cakes cool for 20 minutes in pans; invert onto wire racks and let cool.
Make the meringue: Using an electric hand mixer, beat egg whites in a bowl until medium-stiff peaks form, 5–7 minutes; set aside. Bring sugar and ¼ cup water to a boil in a 2-qt. sauce pan fitted with a candy thermometer; cook, without stirring, until temperature reaches 250°, 4–5 minutes. With the mixer on low, slowly drizzle hot syrup into egg whites until combined. Increase speed to high; beat until meringue forms stiff peaks, about 3 minutes more.
Assemble the cake: Place one cake layer on a cake stand and spread 3 cups meringue over cake; sprinkle with ¼ cup coconut. Repeat with second layer. Place remaining layer on top; cover top and sides with remaining meringue and cover outside of cake with remaining coconut, pressing lightly to adhere. Chill cake 30 minutes to firm meringue; serve chilled or at room temperature.