In a medium bowl, whisk together the flour, baking soda and salt.
In the bowl of a stand mixer fitted with the paddle attachment, cream together the butter and sugars until pale and fluffy, about 5 minutes on medium speed.
Add the eggs, one at a time. Mix in the peanut butter and vanilla, scraping down the sides of the bowl as needed, and continue mixing until well combined.
With the mixer on low speed, add the flour mixture in 3 parts, mixing until just combined. Add the oats and continue mixing on low speed. Stir in the chocolate chips until just combined.
Using a cookie scoop or spoon, drop 1 inch balls of dough onto a baking sheet lined with parchment or a silicone baking mat, about 2 inches apart.
Bake for about 12-14 minutes, until the edges start to brown and are set. If you bake more than one sheet at a time, rotate halfway through baking.
Cool cookies on the baking sheet for 5 minutes after removing them from the oven. After that, move them to a cooling rack to cool completely. Store in an airtight container for up to a week.