Preheat oven to 350 degrees.
In a mixer, cream together butter and sugars until blended. Add vanilla, egg, and banana.
Slowly add flour into mixture until incorporated.
Gently fold in walnuts until blended.
Spoon about 3 tbsp of mixture per cookie onto a greased cookie sheet. Bake at 350 degrees for 18-22 minutes, until tops of cookies start to turn golden brown.