In a large bowl or bowl of a stand mixer, cream butter, white sugar, and brown sugar until smooth. Don't skimp on the mixing, at least 2-3 minutes. Add the eggs and yolk, mix after each addition. Add the vanilla and mix until fluffy. Stir in peanut butter.
In a separate bowl sift together the flour, baking soda, and salt; stir into the peanut butter mixture until combined. Refrigerate the dough for at least 2 hours.
Preheat oven to 350 degrees F. Place parchment or Silpat on cookie sheet.
Roll dough into balls, using 1.5 tablespoons of dough. Place on the prepared cookie sheet and flatten slightly with a fork. Bake for 12 to 15 minutes in the preheated oven. Cookies should look dry on top and golden brown on the bottom edges. Allow to cool for a few minutes on the cookie sheet before removing to cool completely on a rack.