Preheat the oven to 350F and line two baking sheets with parchment paper.
Pulse half the rolled oats in a food processor until they have a more finely ground texture. (optional)
In a large mixing bowl, beat together the butter, brown and white sugars and peanut butter with a mixer until fluffy.
Add the egg and vanilla extract and beat until combined.
With the mixer on low speed, slowly mix in the flour, baking soda and salt.
Add all of the oats to the bowl and stir with a spoon until combined.
Roll the dough into balls that are about 1.5 inches wide. Place them on the parchment lined baking sheet, about 2 inches apart.
Bake the cookies at 350F for 10-12 minutes until the cookies look almost flat (they'll still be puffy), and the tops look set. Don't wait for them to be golden or the cookies won't be soft.
Remove from the oven and let them cool for 10-15 minutes before taking them off the baking sheet.