Preheat an oven to 350 degrees F (175 degrees C). Grease an 11x17 1/2-inch jelly roll pan. Combine white sugar, flour, baking soda, and salt together in a bowl.
Place 1/2 cup butter, shortening, 1/3 cup cocoa powder, and coffee in a microwave safe bowl. Heat in microwave for three minutes, stirring after each minute, until well blended. Whisk together the eggs and buttermilk in a bowl. Stir in the flour and cocoa mixtures, mixing until just combined. Pour mixture into the prepared pan.
Bake in the preheated oven until the top is dry and the edges have started to pull away from the sides of the pan, about 20 minutes.
Heat the remaining 1/2 cup butter, 2 tablespoons cocoa, and milk in a saucepan over medium heat until combined, about 5 minutes, stirring often. Stir in the confectioners' sugar and vanilla extract until smooth. Pour the warm frosting over the hot brownies. Allow brownies to cool completely before slicing into bars.