Place 4 egg whites in a large bowl; let stand at room temperature 30 minutes. Line bottoms of 3 greased 9-in. round baking pans with parchment; grease parchment. Preheat oven to 350°.
Cream butter and 1 cup sugar until light and fluffy, 5-7 minutes; beat in extracts. In another bowl, whisk together flour, baking powder and salt; add to creamed mixture alternately with milk.
With clean beaters, beat the 4 egg whites on medium speed until soft peaks form. Gradually add remaining sugar, 1 tablespoon at a time, beating on high after each addition until sugar is dissolved. Continue beating until stiff glossy peaks form. Fold into batter.
Transfer to prepared pans. Bake until a toothpick inserted in center comes out clean, 13-17 minutes. Cool in pans 10 minutes before removing to wire racks; remove paper. Cool completely.
For frosting, in a large heatproof bowl, whisk 5 egg whites, sugar, water and cream of tartar until blended. Place over simmering water in a large saucepan over medium heat; whisking constantly, heat mixture until a thermometer reads 160°, 2-3 minutes. Remove from heat; add vanilla. Beat on high speed until stiff glossy peaks form, about 7 minutes.
Spread between layers and over top and sides of cake. Cover with coconut. If desired, decorate with sprinkles. Store, uncovered, in refrigerator.
Look for unsweetened coconut in the baking or health food section.