Line two baking sheets with parchment paper.  Set aside.

In the bowl of your electric mixer (or with a hand mixer), cream the butter and sugars until light and fluffy (about 2 - 3 minutes).  Beat in the peanut butter.  Add the egg and vanilla extract and beat to combine.  In a separate bowl whisk together the flour, baking soda, and salt.  Add to the peanut butter mixture and beat until incorporated. 

Cover and chill the dough for about an hour.  Then, on a lightly floured surface, roll the dough to about a 1/4 inch thickness.  Using a lightly floured 3 inch (7.5 cm) cookie cutter, cut out the cookies.  Transfer the cookies to the prepared baking sheet, placing about 2 inches (5 cm) apart.  Then, using the end of a toothpick, score the tops of the cookies, making a crisscross pattern.  Place the baking sheet, with the cookies, in the refrigerator to firm up for about 10 minutes.  

Preheat oven to 350 degrees F (177 degrees C) and place rack in center of oven. 

Bake the cookies for about 12 to 14 minutes, or until the cookies are lightly browned around the edges.  Immediately upon removing the cookies from the oven, sprinkle the tops of the cookies with a little granulated sugar.  Cool on a wire rack. 

Makes about 2 dozen cookies.