Cream together the butter, sugar and vanilla; then beat in the egg.
Combine the flour, baking soda and baking powder and slowly add to the egg mixture and mix well.
Stir in the nuts and make sure they are fully incorporated. Form the dough into a log and wrap well in waxed paper.
Refrigerate until firm (2-4 hours). Preheat the oven to 325 degrees and slice the cookies into 1/4 inch rounds. Place on a non-greased cookie sheet and bake 7-10 minutes until lightly brown on the edges.