1. Preheat the oven to 180C/350F/gas mark 4 and lightly grease a baking tray.
2. Cream together the butter or shortening with the sugar until smooth. Beat in the eggs, vanilla and banana.
3. Combine the flour, baking soda and salt and blend thoroughly into the butter/shortening mixture to make a sticky batter.
4. Drop, one rounded tablespoon at a time, on to the greased baking sheet and bake for 1015 minutes or until lightly browned.
5. Meanwhile, make the sauce by blending together the butter, icing sugar, milk and vanilla. Add more milk as necessary to attain a drizzling consistency. Warm and drizzle over the cooled cookies.