Minetta Taverns Coconut Cake

Name: Minetta Tavern's Coconut Cake
Ingredients: Cake:; 4 ounces (1 stick) unsalted butter, softened; 1/2 cup shortening; 2 1/4 cups sugar4 large eggs, room temperature; 2 cups loosely packed sweetened coconut (about 5 1/4 ounces); 3 cups cake flour, sifted; 4 teaspoons baking powder.; 1/4 teaspoon salt; 1 teaspoon vanilla; 3/4 cup + 1 tablespoon whole milk; Pastry cream:; 1 large egg; 2 large egg yolks; 1/2 cup sugar; 1/4 cup cornstarch; 1 cup whole milk; 1 cup creme fraiche; 1/2 vanilla bean, seeds scraped out and reserved; 2 tablespoons unsalted butter; 1/3 cup heavy whipping cream; Butter cream:; 1/2 cup egg whites (about 4 large eggs); 1/2 cup + 1/3 cup sugar; 10 ounces unsalted butter, softened; 1/2 teaspoon vanilla; Assembly:; 3/4 cup sugar; 3/4 cup water; 4 to 6 tablespoons dark rum; 2 cups toasted unsweetened coconut shavings (about 4 ounces)