Sift together the flour, baking soda, and baking powder into a large bowl, then stir in the oats and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat together the butter, chunky peanut butter, granulated sugar, and brown sugar at medium-high speed until the mixture lightens and fluffs, increasing substantially in volume (by about double), about 5 minutes. Scrape the bowl down as needed (you'll need to often for everything to become cohesive).
Add the eggs, one at a time, beating on medium-low and scraping the bowl down as needed. Add the vanilla extract and beat in until fully incorporated.
Turn the mixer up to medium-high speed and beat until the mixture is smooth, about 30 seconds. Add the dry ingredients all at once, pulse on low speed a few times to avoid spraying flour everywhere, then continue beating at the lowest speed until the dry ingredients are just combined.
Using a 1-ounce cookie scoop (the black-handled scooper), an ice cream scoop, or two spoons, portion out all the cookies onto parchment-lined baking sheets or into Ziploc bags. You should have about 48 cookies total. Place the portioned cookies into your freezer to chill for at least 4 hours or up to overnight (if chilling longer, definitely keep them in a bag, not exposed on the sheet).
Preheat the oven to 375°F. Line two baking sheets with parchment paper and place 8 cookies evenly spaced and staggered onto each sheet. (Note, do not flatten the cookies). Place the baking sheets inside another baking sheet to double-pan them (so you don't burn the outside & bottom of the cookies) and then place the doubled-pans in the oven and bake until evenly golden, about 12 minutes, rotating the pan halfway through baking time (and switching which pan is on which rack if they are on different levels).
Remove from the oven, cool on the sheet for 10 minutes, then transfer to a wire cooling rack to cool completely.