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.
Find dried peaches at Sprouts Farmers Market and Whole Foods Market.