Coconut Layer Cake

1. FOR THE CAKE: Adjust oven rack to lower-middle position and heat oven to 325 degrees. Grease two 9-inch round cake pans, line with parchment paper, grease parchment, and flour pans.
2. Whisk egg and egg whites together in 2-cup liquid measuring cup. Add cream of coconut, water, vanilla, and coconut extract and whisk until thoroughly combined.
3. Using stand mixer fitted with paddle, mix flour, sugar, baking powder, and salt on low speed until combined. Add butter, 1 piece at a time, and mix until only pea-size pieces remain, about 1 minute. Add half of egg mixture, increase speed to medium-high, and beat until light and fluffy, about 1 minute. Reduce speed to medium-low, add remaining milk mixture, and beat until incorporated, about 30 seconds. Give batter final stir by hand.
4. Divide batter evenly between prepared pans and smooth tops with rubber spatula. Bake cake until toothpick inserted in centers comes out clean, about 30 minutes. Do not turn off oven. Let cakes cool in pans on wire rack for 10 minutes. Remove cakes from pans, discard parchment, and let cool completely, about 2 hours, before frosting. (Cooled cakes can be wrapped tightly in plastic wrap and kept at room temperature for up to 1 day. Wrapped tightly in plastic, then aluminum foil, cakes can be frozen for up to 1 month. Defrost cakes at room temperature before unwrapping and frosting.)
5. While cakes are cooling, spread shredded coconut on rimmed baking sheet; toast in oven until shreds are a mix of golden brown and white, about 15 to 20 minutes, stirring 2 or 3 times. Let cool to room temperature.
6. FOR THE ICING: Combine egg whites, sugar, and salt in bowl of stand mixer and set over medium saucepan filled with 1 inch of barely simmering water (do not let bottom of bowl touch water). Whisk constantly until mixture is opaque and warm to the touch and registers about 120 degrees, about 2 minutes.
7. Remove bowl from heat. Fit stand mixer with whisk and whip egg white mixture on high speed until barely warm (about 80 degrees), glossy, and sticky, about 7 minutes. Reduce speed to medium-high and whip in butter, 1 piece at a time, followed by cream of coconut, coconut extract, and vanilla, scraping down bowl as needed. Continue to whip at medium-high speed until combined, about 1 minute.
8. TO ASSEMBLE THE CAKE: Place cooled cake layers on top of each other and make 1⁄8-inch-deep vertical cut into side of each cake layer. With long serrated knife, use sawing motion to cut cakes in half horizontally so that each cake forms 2 layers.
9. Line edges of cake platter with 4 strips of parchment paper to keep platter clean. Place 1 cake layer on prepared platter. Place about 3⁄4 cup icing in center of bottom cake layer and, using large spatula, spread in even layer right to edge of cake. Carefully place other cake layer on top of icing, aligning cuts so that layers are even. Repeat using remaining cake layers and more icing, spreading icing until slightly over edge of top layer. Gather more frosting on tip of spatula and gently spread icing onto side of cake. Smooth icing by gently running edge of spatula around cake and leveling ridge that forms around top edge, or create billows by pressing back of spoon into icing and twirling spoon as you lift away. Carefully pull out pieces of parchment from beneath cake before serving. (Assembled cake can be refrigerated for up to 1 day. Bring to room temperature before serving.)

ASSEMBLING A FOUR-LAYER CAKE
To create an elegant four-layered cake, you must split two cake layers in half horizontally. If you cut the layers a bit unevenly (which is bound to happen), the cake can lean to one side. Here’s how to compensate for less-than-perfect cutting.
1. Place cooled cake layers on top of each other and make 1⁄8-inch-deep vertical cut into side of each cake layer.
2. With long serrated knife, use sawing motion to cut cakes in half horizontally so that each cake forms 2 layers.
3. Assemble cake, aligning cuts in each layer. Stacking layers in their original orientation conceals uneven cutting.