Preheat the oven to 350°F (175°C). Line three baking sheets with parchment paper or silicone mats.
In a large bowl or the bowl of a stand mixer, beat the softened butter, peanut butter, brown sugar, and granulated sugar until well combined and fluffy, about 30 seconds to 1 minute.
Add the eggs and vanilla extract and beat until fully incorporated.
Stir in the flour, quick-cooking oats, baking soda, and salt until just combined and a sticky dough forms; use a wooden spoon if mixing by hand.
Scoop golf-ball-sized portions of dough (about 1.5 tbsp–2 tbsp) onto the prepared baking sheets, spacing about 8 per sheet so cookies can spread.
Bake for 12–15 minutes, until edges are browned and tops look mostly set; bake a bit longer for crisper cookies.
Let cookies cool on the baking sheet until firm enough to transfer, then move to a cooling rack to finish cooling.