Preheat oven to 350 degrees. Stir together oats, flour, baking soda, baking powder, and salt in a medium bowl; set aside.
Put sugars, butter, and peanut butter in the bowl of an electric mixer fitted with the paddle attachment. Mix on medium speed until pale and fluffy, about 5 minutes. Mix in eggs and vanilla.
Reduce speed to low. Add oat mixture, and mix until just combined. Mix in peanuts and chocolate chips.
Using a 1 1/2-inch ice cream scoop, drop balls of dough 2 inches apart on baking sheets lined with parchment paper.
Bake cookies, rotating sheets halfway through, until golden brown and just set, 13 to 15 minutes. Let cool on sheets on wire racks 5 minutes. Transfer cookies to wire racks to cool completely. Cookies can be stored in airtight containers at room temperature up to 2 days.