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3 1/2 cups cake flour, sifted twice
4 tsp. baking powder
1/2 tsp. salt
16 tbsp. (2 sticks/8 oz/226g) unsalted butter
3 1/2 cups granulated sugar
1 cup milk
2 tsp. vanilla extract
1/2 tsp. coconut extract
10 egg whites
5 tbsp. cold water
1/2 tsp. cream of tartar
1 to 1/2 tsp. light corn syrup
2 1/2 cups coarsely grated fresh coconut
Have all ingredients at room temperature. Position a rack in the center of an oven and preheat to 350 degrees F (180 degrees C).
Butter and flour two 9-inch round cake pans. In a large bowl, sift together the flour, baking powder and salt. In the bowl of an electric mixer, beat the butter on high speed until creamy, 4 to 5 minutes. Gradually add 2 cups of the sugar and continue beating, scraping down the sides of the bowl, until light and fluffy, 4 to 5 minutes more. Reduce speed to low. Fold the flour mixture into the butter in three parts, alternating with the milk, until smooth. Fold in 1 tsp. of the vanilla and the coconut extract.
In a large, clean bowl, beat 8 of the egg whites with a whisk to stiff peaks. Fold 1/2 cup whites into the batter until blended, then fold in the remaining whites. Divide the batter among the prepared pans. Bake until golden and a toothpick inserted into the centers comes out clean, 30 to 35 minutes. Cool to room temperature.
In a heatproof bowl, combine the remaining 2 egg whites, the remaining 1 1/2 cups sugar, the water, cream of tartar and corn syrup. In a bowl set over a pan of simmering water but not touching the water, beat the mixture until thick, shiny and double in volume, about 7 minutes. Remove from heat; add the remaining 1 tsp. vanilla and beat until the icing is thick enough to spread, 2 to 3 minutes more.
In a small bowl, combine 1 cup icing and 1 1/2 cups of the coconut; stir to mix. Set 1 cake on a plate, spread the icing evenly over the top and set the other cake on top. Working from top to bottom, spread on the remaining plain icing. Pat the remaining coconut evenly onto the sides and top.