Adjust oven rack to the middle position and preheat oven to 350°F. Spray a 9x13-inch baking dish or 3 (8-inch) round cake pans with baking spray. Set aside.
Separate the eggs (yolks and whites) into separate bowls and use a hand mixer to beat the egg whites just to stiff peaks. Set the beaten whites and yolks aside.
5 large eggs
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and shortening together at medium speed until creamy and smooth, about 2 minutes.
½ cup unsalted butter, ½ cup vegetable shortening
Add the sugar and beat until lighter in color and fluffier, about 3 minutes.
2 cups granulated sugar
Add the egg yolks, beating until incorporated.
In a medium bowl, whisk the flour and salt together.
2 cups all-purpose flour, 1 teaspoon kosher salt
In a separate medium bowl, dissolve the baking soda in the buttermilk.
1 cup buttermilk, 1¼ teaspoons baking soda
Add the flour mixture to butter/sugar/egg, mixture alternately with the buttermilk/baking soda mixture in 3 additions; beginning and ending with the flour mixture. Beat just until incorporated after each addition.
Stir in the vanilla and coconut extracts, toasted pecans, and coconut.
1 teaspoon pure vanilla extract, 1 teaspoon coconut extract, 1 cup sweetened shredded coconut, 1 cup chopped pecans
Fold in the beaten egg whites.
Pour the batter into the prepared pan(s). For a 9x13-inch cake, bake 30-40 minutes, or until a toothpick inserted in the middle comes out clean. For 8-inch round cakes, bake 25-30 minutes.
Cool the 9x13 cake in the pan set on a wire rack. If baking round cakes, remove them from the pans after 10 minutes. Cool completely on the wire racks, about 2 hours.