Preheat oven to 350 degrees F (175 degrees C). Lightly grease baking sheets.
In a medium bowl, cream together shortening and white sugar until smooth. Beat in eggs, vanilla extract, and banana. Combine flour, baking soda, and salt; blend thoroughly into the shortening mixture to make a sticky batter. Drop by rounded tablespoons onto the prepared baking sheets.
Bake 10 to 15 minutes in the preheated oven, or until lightly browned.
In a medium bowl, blend butter, confectioners' sugar, milk and vanilla extract. Adjust amount of milk as necessary to attain a drizzling consistency. Drizzle over warm cookies.