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1. For the Cake: Adjust oven rack to lower-middle position and heat oven to 325 degrees. Grease two 9-inch round cake pans with shortening and dust with flour.
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2. Beat egg whites and whole egg in large measuring cup with fork to combine. Add cream of coconut, water, vanilla, and coconut extract and beat with fork until thoroughly combined.
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3. Combine flour, sugar, baking powder, and salt in bowl of standing mixer fitted with paddle attachment. Mix on lowest speed to combine, about 30 seconds. With mixer still running on lowest speed, add butter 1 piece at a time, then beat until mixture resembles coarse meal, with butter bits no larger than small peas, 2 to 2 1/2 minutes.
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4. With mixer still running, add 1 cup liquid. Increase speed to medium-high and beat until light and fluffy, about 45 seconds. With mixer still running, add remaining 1 cup liquid in steady stream (this should take about 15 seconds). Stop mixer and scrape down bowl with rubber spatula, then beat at medium-high speed to combine, about 15 seconds. (Batter will be thick.).
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5. Divide batter between cake pans and level with offset or rubber spatula. Bake until deep golden brown, cakes pull away from sides of pans, and toothpick inserted into center of cakes comes out clean, about 30 minutes (rotate cakes after about 20 minutes). Do not turn off oven.
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6. Cool in pans on wire racks about 10 minutes, then loosen cakes from sides of pans with paring knife, invert cakes onto racks and then re-invert; cool to room temperature.
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7. 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. Cool to room temperature.
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8. For the Buttercream: Combine whites, sugar, and salt in bowl of standing mixer; set bowl over saucepan containing 1 1/2-inches of barely simmering water. Whisk constantly until mixture is opaque and warm to the touch and registers about 120 degrees on an instant-read thermometer, about 2 minutes.
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9. Transfer bowl to mixer and beat whites on high speed with whisk attachment until barely warm (about 80 degrees) and whites are glossy and sticky, about 7 minutes. Reduce speed to medium-high and beat in butter 1 piece at a time. Beat in cream of coconut and coconut and vanilla extracts. Stop mixer and scrape bottom and sides of bowl. Continue to beat at medium-high speed until well-combined, about 1 minute.
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10. To Assemble the Cake: Follow illustrations in chart below. Cut into slices and serve. (Wrap leftover cake in plastic and refrigerate; bring to room temperature before serving.).
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ASSEMBLING THE CAKE.
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1. With a long serrated knife, cut both cakes in half horizontally so that each cake forms two layers.
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2. Put a dab of icing on a cardboard round cut just larger than the cake. Center one cake layer on the round.
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3. Place a large blob of icing in the center of the layer and spread it to the edges with an icing spatula.
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4. Hold the spatula at a 45-degree angle to the cake and drag it across the surface to level the icing. Repeat steps 3 and 4 with remaining cake layers.
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5. To ice the sides of the cake, scoop up a large dab of icing with the tip of the spatula and spread it on the sides with short side-to-side strokes.
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6. Sprinkle the top of the cake with coconut. Then press the coconut into the sides, letting the excess fall back onto a baking sheet.
