How to Make It
Heat oven to 350° F. Butter an 11-by-17-inch rimmed baking sheet and line with parchment; butter the parchment and dust lightly with flour. Whisk together the flour, baking powder, baking soda, and salt in a medium bowl. In a separate bowl, whisk together the egg whites, milk, and vanilla.
Beat the butter and sugar in a large bowl with an electric mixer on medium-high until light and fluffy, 2 to 3 minutes.
Reduce mixer speed to low; add the flour mixture and milk mixture in alternate additions, beginning and ending with the flour mixture. Mix until just combined.
Transfer the batter to the prepared sheet. Bake until the cake springs back lightly when pressed, 15 to 20 minutes. Cool, then transfer to a cutting board.
Using a long, serrated knife, cut the cake into 4 pieces of equal size, slicing crosswise and lengthwise through the center.
Place one layer of the cake on a serving platter and frost the top with ⅔ cup of the frosting. Repeat with 2 of the remaining layers. Top with the final layer. Frost the top and sides of the cake with the remaining 2 cups of frosting. Press the coconut into the top and sides of the cake.