Preheat oven to 350°F.
Grease and flour three 8-inch round cake pans, then line with parchment paper. Set the pans aside until ready to use.
Put on a pair of vinyl glove and using your hands, separate the eggs. Allow them to come to room temperature.
Measure one cup of buttermilk in a two-cup measuring cup. Add the baking soda and stir to combine. (The chemical reaction between the two will cause the buttermilk to double in volume.) Set aside until ready to use.
Whisk together the cake flour and table salt. Set aside until ready to use.
Place the unsalted butter, shortening, and white sugar in the bowl of a stand mixer. Fit it with the paddle attachment and mix on medium-high speed for 3-5 minutes.
Turn the mixer off and scrape down the inside of the bowl. Then, with the machine running on medium, add the egg yolks one at a time. Add the vanilla extract.
Add the cake flour mixture and buttermilk mixture alternately, beginning with the cake flour, about ½ cup at a time, and ending with the buttermilk.
Transfer the batter to a large mixing bowl. Add the toasted pecan pieces and sweetened coconut flakes. Gently fold them into the batter.
Wash and dry the bowl from the stand mixer really well. (If the bowl is not clean and dry, the egg whites won't whip properly.) Add the egg white to the bowl, then fit the mixer with the whisk attachment. Turn the machine on low, then gradually increase the mixer speed to high and whisk until stiff peaks are formed.
Add about ⅓ of the whipped egg whites to the batter, then gently fold them in. Add the remaining egg whites and gently fold them into the batter.
Using a large ice cream scoop evenly divide the batter between the prepared pans. Using a silicone spatula, carefully smooth the batter out evenly in the pans.
Bake all three pans in a 350°F until a toothpick inserted in the center comes out clean, for 20-25 minutes, rotating the pans halfway through baking.
Remove the pans from the oven and transfer them to a wire cooling rack. Let the cakes cool in the pans for 10 minutes. Then, carefully remove them from the pans and allow them to cool completely, right side up, on cooling racks. If the cakes are upside down, the tops will stick to the racks and damage them.