Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper.
In a medium bowl, combine flour, baking soda, and salt. Set aside.
In a large bowl, cream together the butter, granulated sugar, and brown sugar until smooth.
Stir in the peanut butter and vanilla. Beat in the eggs one at a time.
Gradually blend in the flour mixture. Stir in the oats until just combined.
Drop by rounded tablespoonfuls onto the prepared cookie sheets.
Bake for 10 to 12 minutes in the preheated oven, or until edges are golden. Cool on baking sheets briefly before transferring to wire racks to cool completely.
Notes
These cookies can be stored in an airtight container for up to one week.