In a medium bowl, combine the flour, baking soda and salt.
In a large bowl, using an electric mixer cream the butter, peanut butter and sugars until light and fluffy. Beat in the egg and vanilla extract. Fold the dry ingredients into the wet ingredients. Cover the bowl with plastic wrap and chill the dough in the refrigerator overnight.
Preheat the oven to 350 degrees F. Line a sheet pan with parchment. Drop the dough by the scant quarter cup full and place 2 inches apart on the prepared sheet pan.
Using a fork, press a crisscross pattern into the dough. Bake the cookies for 12-15 minutes until they are golden brown. Cool for 5 minutes on the sheet pan before transferring to a wire rack to cool completely.