Preheat the oven to 350 degrees F.

FOR THE CAKE: Grease a 1/2 sheetpan (a 17 by 12-inch rimmed baking sheet) with 1 tablespoon of butter. Sift the flour, cocoa, soda, and salt into a large mixing bowl. In a saucepan, over medium heat, combine the butter, shortening, sugar, and water. Bring to a boil. Stir the liquid mixture into the sifted mixture. Mix well. Add the eggs, buttermilk and vanilla. Mix well. Spread evenly into the prepared sheet pan. Bake for 25 minutes. Remove from the oven.

FOR THE ICING: In another saucepan, over medium heat, combine the butter and milk. Bring to a boil. Sift the confectioners' sugar and cocoa powder into a large bowl. Add the butter mixture and whisk well. Add the vanilla. Pour the icing over the warm cake. Sprinkle the pecans over the frosting. Allow the cake to sit until the frosting is set, about 15 minutes.

To serve, place a slice of the cake in the center of each serving plate. Place a scoop of the ice cream in the center of each slice of cake.