Preheat the oven to 350 F. In a medium bowl, whisk together flour, oats, salt, and baking soda. In a large bowl, combine butter with peanut butter, and both sugars. Mix until fluffy.
Add in egg and vanilla extract, and mix until smooth. Add in the flour mixture, and mix until a thick dough forms. Roll the cookie dough into balls, and set them onto a large ungreased baking sheet.