Preheat the oven to 350°F (175°C) and grease three 9-inch round cake pans.
Make the Cake Batter: In a large mixing bowl, beat together 2 cups of sugar, 6 large eggs, and 6 egg yolks using an electric mixer until light and fluffy, about 5 minutes.
Heat the Milk: In a saucepan, heat 1 cup of milk and 1 cup of unsalted butter over low heat until the butter melts, 2 to 3 minutes. Remove from heat and let it cool slightly.
Combine Dry Ingredients: In a separate bowl, mix together 3 cups of all-purpose flour and 3 teaspoons of baking powder. Pour the flour mixture into the egg mixture. Add 1 teaspoon of vanilla extract and the milk mixture. Beat well using an electric mixer. Divide the batter evenly among the prepared cake pans.
Bake the Cakes: Place the pans in the preheated oven and bake until a toothpick inserted into the center of the cakes comes out clean, 15 to 18 minutes. Let the cakes cool in the pans for 10 minutes before transferring them to wire racks to cool completely.
Make the Meringue: Thoroughly clean the mixing bowl and beaters. Beat 3 egg whites and 1/2 teaspoon of cream of tartar with an electric mixer until foamy. In a saucepan, bring 1 cup of sugar and 1/2 cup of water to a boil. Boil until a candy thermometer reads 242°F (116°C), 3 to 4 minutes.
Finish the Meringue: Beat the egg whites on high speed. Carefully pour in the hot sugar mixture along the side of the bowl, avoiding the beaters. Continue to beat on high until stiff peaks form, 6 to 8 minutes. Stir in 1 teaspoon of vanilla extract.
Assemble the Cake: Place one cooled cake layer on a serving platter and frost the top. Repeat with the second and third layers of cake. Completely frost the sides, top, and throughout. Sprinkle 2 cups of shredded coconut on top and smooth it onto the sides of the cake using your hands.
