In a large mixing bowl, cream together butter, sugar and brown sugar until light and fluffy, about 2-3 minutes. If your butter is very soft you can do this with a spoon. Otherwise, you can use an electric mixer.
Add egg and vanilla. Mix until smooth, scraping down sides of bowl as necessary. Stir in peanut butter until smooth.
In a smaller mixing bowl, whisk together all purpose flour, baking soda and salt. Stir into the creamed peanut butter mixture in 2 or 3 stages, mixing until smooth. Dough will be very soft. Cover with plastic and refrigerate at least 1 hour or until firm enough to roll into balls.
Preheat oven to 350℉/177℃. Roll dough into walnut sized balls and place onto parchment lined baking sheets. Using the tines of a fork, press a crisscross pattern into each ball, flattening down only slightly. You should be able to fit 12 cookies onto each tray, spacing them 1½ inches apart.
Bake for 16-18 minutes until tops of cookies are barely shiny and bottoms are lightly golden. The longer you bake them, the crunchier they will be. Cool 2-3 minutes on baking sheets before transferring onto wire racks to cool completely. Store in an airtight container or cookie jar for up to 5 days.