How to Make It
Prepare Cake Layers: Preheat oven to 350°. Beat first 2 ingredients at medium speed with an electric mixer until fluffy. Gradually add 8 egg whites, 2 at a time, beating well after each addition.
Sift together flour and baking powder; gradually add to butter mixture alternately with 1 cup water, beginning and ending with flour mixture. Stir in 1 Tbsp. vanilla extract. Spoon batter into 4 greased (with shortening) and floured 9-inch round shiny cake pans (about 1 3/4 cups batter in each pan).
Bake at 350° for 14 to 16 minutes or until a wooden pick inserted in center comes out clean. Cool in pans on wire racks 10 minutes; remove from pans to wire racks, and cool completely (about 30 minutes).
Prepare Filling: Pour boiling water to cover over dried peach halves in a medium bowl; let stand 30 minutes. Drain well, and cut into 1/4-inch pieces. (After plumping and dicing, you should have about 2 cups peaches.)
Whisk together melted butter and next 2 ingredients in a heavy saucepan. Cook over medium-low heat, whisking constantly, 10 to 12 minutes or until thickened. Remove from heat, and stir in diced peaches, coconut, and next 3 ingredients. Cool completely (about 30 minutes).
Spread filling between cake layers (a little over 1 cup per layer). Cover cake with plastic wrap, and chill 12 hours.
Prepare Frosting: Pour water to a depth of 1 1/2 inches into a small saucepan; bring to a boil over medium heat. Whisk together 2 egg whites, 1 1/2 cups sugar, and next 3 ingredients in a heatproof bowl; place bowl over boiling water. Beat egg white mixture at medium-high speed with a handheld electric mixer 12 to 15 minutes or until stiff glossy peaks form and frosting is spreading consistency. Remove from heat, and spread immediately over top and sides of cake.