How to Make It
Prepare the Cake: Preheat oven to 350°F. Combine boiling water, cocoa, unsweetened chocolate, and, if desired, instant coffee in a bowl. Let stand 2 minutes; whisk until smooth. Let cool completely.
Whisk together flour, baking powder, and salt in a bowl. Beat butter with an electric mixer at medium speed until smooth, about 1 minute. Add brown sugar and 1 cup granulated sugar; beat until light and fluffy, about 2 minutes. Add eggs, 1 at a time, beating well after each addition. Beat in sour cream and 1 teaspoon vanilla. Add flour mixture to butter mixture alternately with cocoa mixture, in 5 additions, beginning and ending with flour mixture. Beat at medium speed until smooth, about 1 minute. Divide batter evenly between 2 (9-inch) greased and floured round cake pans. Bake in preheated oven until a wooden pick inserted in the center comes out clean, 28 to 32 minutes. Cool in pans on a wire rack 10 minutes. Remove cake from pans; cool completely, about 30 minutes.
Prepare the Coconut Cream Filling: Stir together half-and-half and shredded coconut in a small saucepan over medium; bring just to a boil. Remove pan from heat; let stand 30 minutes. Pour mixture through a fine wire-mesh strainer into a bowl, pressing on solids; discard solids. Return coconut liquid to pan; bring to a simmer. (Coconut liquid should just begin to bubble around the edges of pan.) Whisk together 5 tablespoons granulated sugar, cornstarch, and egg yolks in a medium bowl until smooth. Gradually add hot coconut liquid to egg mixture, whisking constantly. Return mixture to pan. Cook, whisking constantly, over medium heat until thick and bubbly, 5 to 6 minutes. Remove pan from heat. Whisk in 2 tablespoons butter, rum, 1 teaspoon vanilla, and coconut extract. Pour mixture into a shallow dish; refrigerate until completely cold, about 2 hours. Transfer chilled custard to a medium bowl.
Beat 1/2 cup heavy cream until stiff peaks form. Gently fold whipped cream into custard in 2 batches. Refrigerate until ready to use.
Prepare Chocolate Frosting: Combine 2 cups heavy cream and powdered sugar in a medium saucepan. Bring to a simmer over medium. (Do not boil.) Place bittersweet chocolate in a medium bowl; pour hot cream mixture over chocolate. Let stand 1 minute; gently stir until all of the chocolate melts and mixture is smooth. Chill until mixture thickens to spreading consistency, 30 to 40 minutes, stirring every few minutes.
To assemble cake, slice cake layers in half horizontally using a serrated knife; place 1 layer, cut side up, on a plate. Spread about 1/4 cup Chocolate Frosting over top. Top with one-third of Coconut Cream Filling, leaving a 1/2-inch border. Top with another cake layer; repeat layers twice, beginning with frosting and ending with a cake layer. Spread remaining Chocolate Frosting on top and sides of cake. (If chocolate gets too thick to spread, microwave at HIGH until spreadable, 5 to 7 minutes.) Press unsweetened flaked coconut around sides of cake.