Have all ingredients at room temp. Position rack in center of oven and preheat to 350 degrees F. Butter and flour two 9” round cake pans. Sift together flour, baking powder, and salt. In a large bowl beat butter on high speed until creamy, 4-5 min. Gradually add 2 cups sugar and continue beating, scraping down sides of bowl, until light and fluffy, 4-5 min more. Reduce speed to low. Fold flour mixture into butter in three parts, alternating with milk, until smooth; fold in 1 tsp vanilla and coconut extract. In a clean bowl, beat 8 egg whites with a whisk to stiff peaks. Fold 1/2 cup egg whites into batter until blended, then fold in remaining egg whites. Divide batter among pans; bake until golden, 30-35 min.
In a heatproof bowl, combine remaining 2 egg whites, 1 1/2 cups sugar, water, cream of tartar and corn syrup. Set bowl over simmering water, but not touching water, beat mixture until thick, shiny and double in volume, about 7 min. Remove from heat; add 1 tsp vanilla and beat until thick enough to spread, 2-3 min.
In a small bowl, combine 1 cup icing and 1 1/2 cups coconut; stir to mix. Set one cake on a plate, spread icing evenly over top and set other cake on top. Working from top to bottom, spread remaining plain icing. Pat remaining coconut evenly onto sides and top.