Preheat the oven to 350°F and line baking sheets with parchment paper.
Beat the butter, peanut butter, brown sugar, and granulated sugar until creamy and fluffy.
Add the eggs one at a time, then mix in the vanilla extract.
Whisk together the flour, oats, baking soda, and salt. Stir the dry ingredients into the wet mixture just until combined.
Scoop rounded tablespoons of dough onto the prepared sheets, leaving space between each cookie.
Bake for 12 to 15 minutes, until the edges are golden and the centers look slightly soft.
Cool the cookies on the baking sheet for 10 minutes, then transfer them to a wire rack.