In a large bowl, cream together the butter, white sugar, and brown sugar until light and fluffy. Beat in the egg, then stir in the vanilla and peanut butter. Combine the flour, baking soda and salt; stir into the peanut butter mixture. Finally, fold in the chopped peanuts. Roll dough into walnut sized balls, place onto an unprepared cookie sheet, and press down slightly with a fork.
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.