Make a banana cake with Pillsbury cake mix by mixing a mashed large banana, 3 eggs and a cup of water with the cake mix. Bake the cake in an oven preheated to 350 degrees Fahrenheit for 30 to 35 minutes, cool and frost the cake.
Before mixing the cake, grease a 13- by 9-inch pan. Use a yellow cake mix with pudding in the mix for the recipe. Mix the cake using the package directions but substituting the banana, eggs and water for the ingredients the directions list. The cake is done when you insert a toothpick in the center and it comes out clean. Place the pan on a cooling rack away from the hot oven to cool for about an hour.
Prepare the frosting by melting 1 cup of butter in a saucepan, using medium heat. Continue heating the butter until it begins to turn brown, and remove it from the heat. Once it cools, add 4 cups of powdered sugar and 1 teaspoon of vanilla. Beat the frosting with an electric mixer and add 2 to 4 tablespoons of milk to thin it to the consistency you desire. Spread the frosting on the cool cake and serve it.