Start by beating the softened butter, peanut butter, brown sugar, and granulated sugar together in a large bowl until the mixture is light and fluffy, about 3 to 5 minutes with an electric mixer on medium speed.
Beat in the eggs one at a time, ensuring each is fully incorporated before adding the next. Then, stir in the vanilla extract.
In a separate bowl, whisk together the all-purpose flour, quick-cooking oats, baking soda, and salt.
Gradually add the dry mixture to the wet ingredients, folding gently with a spatula until just combined. Be careful not to overmix.
Using a cookie scoop or tablespoon, drop rounded dough balls onto a parchment-lined baking sheet, spacing them about 2 inches apart.
Bake in a preheated 350°F (175°C) oven for 10 to 12 minutes, or until the edges start to turn golden but centers remain soft.
Allow cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.