Prepare the Cake Layers: Pulse shredded coconut in a food processor until finely chopped, about 20 times. Transfer to a bowl. Heat heavy cream in a small saucepan over medium just until warm and beginning to release steam, about 5 minutes. Pour warmed cream over coconut, and cool completely, about 30 minutes.

Preheat oven to 350°F. Beat granulated sugar and butter with an electric mixer on medium speed until light and fluffy, about 2 minutes. Add egg yolks, 1 at a time, and beat until blended after each addition. Add vanilla, and beat just until smooth. Whisk together flour, baking powder, baking soda, and salt in a medium bowl. Add flour mixture, in batches, alternately with cream mixture, beginning and ending with flour mixture. Beat on low speed until blended after each addition.

Beat egg whites in a bowl with an electric mixer fitted with whisk attachment on high speed until stiff peaks form. Stir about one-third of egg whites into batter; fold in remaining egg whites. Pour batter into 4 greased (with shortening) and floured 8-inch round cake pans.

Bake in preheated oven until a wooden pick inserted in center comes out clean, 25 to 30 minutes. Cool in pans on wire racks 20 minutes; remove from pans to wire racks, and cool completely, about 30 minutes.

Prepare the Filling: Stir together half-and-half and next 3 ingredients in a small saucepan. Cook over medium-high, whisking constantly, just until mixture thickens and comes to a boil, 10 to 12 minutes. Remove from heat, and whisk in butter and rum until smooth. Transfer to a glass bowl, and place plastic wrap directly on warm custard (to keep a film from forming). Cool completely, about 30 minutes. Chill until ready to use.

Prepare the Frosting: Whisk together granulated sugar, milk, flour, and salt in a medium saucepan. Cook over medium, whisking constantly, until mixture thickens, about 10 minutes. Immediately pour mixture into a medium-size metal bowl, and cool completely, about 30 minutes, stirring occasionally.

Beat butter with an electric mixer on medium speed until smooth. Slowly add cooled custard to butter mixture, 1/4 cup at a time, and beat until smooth after each addition. Add coconut extract and vanilla, and beat until smooth. Spoon 1 cup of the Frosting into a ziplock plastic freezer bag, snip off 1 corner for piping.

Assemble Cake: Place 1 Cake Layer on a cake plate, and pipe a 1/2-inch-thick ring of Frosting around the outer edge of the layer. Spread about 2/3 cup of the Filling over the layer. Repeat with second and third Cake Layers. Place the final layer on top, and frost sides and top with remaining Frosting.