To make the peanut butter cookies, in a bowl, sift together the flour, baking soda, and baking powder. Stir in the oats and salt. Set the dry ingredients aside.
In the bowl of an electric mixer with the paddle attachment, combine the butter, chunky peanut butter, and sugars and cream on medium-high speed until very fluffy and pale, at least 3 minutes, scraping down the mixing bowl as needed.
Turn the mixer to medium-low and add the eggs, one at a time, beating to incorporate each egg and scraping down the bowl as needed. Beat in the vanilla extract. Add the dry ingredients on low speed in 3 to 4 additions and mix until just combined. Do not overmix. Remove the bowl from the mixer and scrape down the bowl with a rubber spatula, going all the way to the bottom of the bowl to mix in the dry ingredients well.
Use an ice cream scoop to portion all the cookies in 1-ounce scoops (or use about 1 heaping tablespoon per cookie), placing the scoops on a parchment lined baking sheet. You should have about 48 cookies. (You can place all the cookies close together for the chilling step—you will space them for baking later.) Chill the scooped cookies for at least 2 hours or longer.
When you are ready to bake the cookies, preheat the oven to 375°F. Arrange 8 cookies, spaced evenly apart and staggered, on each parchment-lined baking sheet. (Note: Do not flatten the cookies; they will flatten as they bake.) Set the baking sheet inside another baking sheet to double pan and place it in the oven. Bake until evenly golden, about 12 minutes, rotating the pan halfway through the cooking time. If you have 2 double-panned pans in the oven at the same time, also switch them between the racks. Remove the pan from the oven and cool on a wire rack for about 10 minutes before removing the cookies with a metal spatula. Allow the cookies to cool completely before filling them.