Preheat the oven to 350 degrees F (175 degrees C). Grease three 9-inch round cake pans.
Mix sugar, eggs, and egg yolks together in a mixing bowl using an electric mixer until light and fluffy, about 5 minutes.
Heat milk and butter in saucepan over low heat until butter melts, 2 to 3 minutes. Remove from heat and let cool slightly.
Combine flour and baking powder in a bowl. Pour into egg mixture. Add vanilla extract and milk mixture; beat well using an electric mixer. Divide batter evenly among the prepared pans.
Bake in the preheated oven until a toothpick inserted into the center of the cakes comes out clean, 15 to 18 minutes. Let cool in pans for 10 minutes before transferring to wire racks to cool completely.
Meanwhile, thoroughly clean mixing bowl and beaters. Beat egg whites and cream of tartar with an electric mixer until foamy.
Bring sugar and water to a boil in a saucepan. Boil until a candy thermometer reads 242 degrees F (116 degrees C), 3 to 4 minutes.
Beat egg whites on high speed. Carefully pour in hot sugar mixture in the side of the bowl, avoiding the beaters; continue to beat on high until stiff peaks form, 6 to 8 minutes. Stir in vanilla extract.
Place 1 cooled cake layer on a serving platter and frost the top. Repeat with second and third layers of cake. Complete frosting the sides, top, and throughout. Sprinkle coconut on top and smooth onto sides of cake using your hands.
For best results for the frosting, make sure the bowl and egg beaters are clean with no trace of grease and the egg whites have no streaks of yolk. Room temperature egg whites will hold more volume than cold eggs.
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