In the bowl of a stand mixer, or in a large bowl with an electric mixer (you can mix by hand too, it'll just take some muscle!), combine butter, sugars and peanut butter until smooth. Beat in egg and vanilla until combined.
Add flour, baking soda, salt and oats and mix until just combined (don't overmix). Stir in chocolate chips.
Lightly grease 2 baking sheets or line with parchment paper. Roll cookie dough into 1" balls and place 2" apart on the baking sheet (I do 12 to one baking sheet). Press down slightly (the more you flatten them, the thinner they will end up) and bake for 8-10 minutes until just slightly glossy in the center. Remove from the oven and let cool for 10 minutes before removing to a cooling rack.
Notes
I baked mine exactly 9 minutes and took them out when the centers were still slightly glossy -- I do not like overbaked cookies! They will continue to cook and set up somewhat after they are removed from the oven, so don't be afraid to take them out just before they're set completely.