Prepare the oven and pan. Preheat oven to 350°F. Grease and flour a 9-by-13-by-2-inch baking pan.

Make the batter. In a large mixing bowl or bowl of a stand mixer, combine the flour, granulated sugar, and salt; whisk or stir to blend thoroughly. Set aside.
In a small saucepan, combine 1/2 cup butter, the shortening, 1/4 cup of cocoa, and 1 cup of water; bring to a boil over high heat. Pour the mixture over the flour mixture. Add the buttermilk, baking soda, eggs, and vanilla. Beat with an electric mixer until smooth. Spoon the batter into the prepared baking pan and spread evenly.

Bake the cake. Bake in the preheated oven for 40 to 45 minutes, or until the cake springs back when lightly touched with a finger. The cake is frosted while hot, so make the frosting just before the cake is done.

Frosting

Combine the butter, cocoa, and milk. Combine 1/2 cup of butter, cocoa, and milk in a medium saucepan. Bring the mixture just to a boil over medium-high heat; remove from the heat.

Add the remaining frosting ingredients. Add confectioners’ sugar and vanilla to the saucepan and beat with a wooden spoon until smooth. Stir in coconut and pecans.

Frost the hot cake. Spread the hot cake right out of the oven. Cool the frosted cake on a rack, then slice and enjoy.