Heat oven to 350°F (325°F for dark pan). Grease bottom only of 13x9-inch pan with shortening. Make cake mix as directed on box adding mashed banana with water and eggs. Pour batter into pan.
Bake 30 to 35 minutes or until toothpick inserted in center comes out clean. Cool completely, about 1 hour.
In 3-quart saucepan, melt butter over medium heat. Cook 4 to 6 minutes, stirring constantly and watching closely, until butter just begins to turn golden (butter will get foamy and bubble). Remove from heat. Cool 15 minutes.
With electric mixer on low speed, beat in powdered sugar, vanilla and enough milk until frosting is smooth and desired spreading consistency. Frost cooled cake.