Preheat oven to 350 degrees F (175 degrees C). Grease cookie sheets. Sift together the flour, baking powder and salt, set aside.
In a medium bowl, cream together the butter and brown sugar. Beat in the banana and egg, then stir in the vanilla. Gradually stir in the dry ingredients until well blended. Fold in the chocolate chips and walnuts, if desired. Drop by rounded spoonfuls onto the prepared cookie sheets.
Bake for 8 to 10 minutes in the preheated oven. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.