Preheat oven to 350°F and line three baking sheets with parchment paper or silicone baking mats.
In a large bowl or stand mixer, beat together softened butter, peanut butter, brown sugar, and granulated sugar until light and fluffy, about 30 seconds to 1 minute.
Add eggs one at a time, mixing well after each addition, then stir in vanilla extract until completely combined.
In a separate bowl, whisk together flour, oats, baking soda, and salt. Gradually add dry ingredients to wet mixture, stirring just until combined. Dough will be sticky.
Scoop golf ball-sized portions of dough onto prepared baking sheets, spacing them at least 3 inches apart (cookies spread significantly).
Use the back of a fork dipped in sugar to press a crisscross pattern on top of each cookie to help even spreading.
Bake for 12 to 15 minutes until edges are golden brown and tops look mostly set. Do not overbake.
Cool on baking sheet for at least 10 minutes before transferring to a wire rack to cool completely.