Preheat your oven to 350 degrees and line two baking sheets with parchment paper. In a small bowl, stir together the flour, baking soda, and salt, and set aside. In a large bowl, beat together the butter, brown sugar, and granulated sugar until well combined. Beat in the egg and vanilla extract until just combined. Finally, beat in the dry ingredients and peanut butter slowly until everything is well combined and you have a firm dough.