Preheat oven to 350°F (175°C) and line baking sheets with parchment paper.
In a large bowl, cream together the butter, peanut butter, granulated sugar, and brown sugar until smooth.
Beat in the egg and vanilla extract until well combined.
In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.
Gradually stir the dry ingredients into the peanut butter mixture until just incorporated.
Fold in the rolled oats until evenly distributed throughout the dough.
Drop rounded tablespoonfuls of dough onto the prepared baking sheets, spaced 2 inches apart.
Bake for 10 to 12 minutes or until the edges are lightly browned and the centers are set.
Allow cookies to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.