Preheat oven to 350 degrees. Grease a jelly roll pan and line with greased parchment paper.
Mix chocolate cake batter according to package directions. Spread the batter into prepared jelly roll pan, and bake in a preheated oven for 15-20 minutes. Let cool in pan on a wire rack.
In a saucepan, combine flour and milk. Cook over medium heat until thickened, stirring constantly. Let cool completely.
In a bowl with an electric mixer, beat together 1/2 cup butter, sugar, and shortening. Add this to the flour mixture and beat until fluffy, about 10 minutes. Spread icing on cooled cake. Refrigerate.
In another bowl, beat together 1/2 cup butter, egg, melted chocolate, vanilla, powered sugar, and hot water until smooth and creamy. Carefully spread topping over fluffy icing to create a second layer. Refrigerate until ready to serve. Makes 12 servings. Enjoy!