Preheat oven to 350 F. Line a baking sheet with parchment paper or a silicone baking mat, and set aside.
Whisk together the flour, baking powder, baking soda, cornstarch, and salt in a large bowl. Set aside.
In a stand mixer (or with a hand mixer) with a paddle attachment, beat together the butter, granulated sugar and brown sugar on medium speed until light and fluffy, about 3 minutes.
Add in all of the peanut butter, and mix until it is incorporated and smooth.
Add in the vanilla and eggs, one at a time, mixing between each addition just until they are mixed. If needed, scrape down the sides of the bowl.
Reduce the mixer speed to low, and slowly add the dry ingredients just until they are incorporated and the batter is smooth. Be careful not to over mix the batter.
Using a small batter scoop, scoop the batter onto the baking sheet. Press a fork onto the ball, flattening it into a cookie shape. Be sure that there is at least 2 inches between each cookie during baking.
Bake for 10-12 minutes, or until the edges are golden brown.
Remove the cookies from the oven and let sit for 10 minutes, then transfer the cookies to a cooling rack to completely cool.
I use a small batter scoop to ensure that each cookie is perfectly the same size, and to make the job of moving around the batter much easier. I also use the silicone baking sheets instead of regular parchment paper, since I find them easier to work with, and they are easily cleaned with soapy water.
These cookies will stay good packed in an air-tight container for 4 days, or they can be frozen. To freeze them, just let them cool after baking, and seal them in a air-tight bag and put it into the freezer. The frozen cookies will stay good for 3 months, and are perfect for when you have unexpected guests, or you want to stockpile your cookie stash. To reheat them, you can just pop them in the oven at 350F, or just put them into a toaster oven for 10 minutes.
For the type of peanut butter, there is no real difference in the recipe whether you use smooth or chunky peanut butter, however, it does change the consistency a bit. While using chunky peanut butter will add peanuts into the cookie, it might be preferred to use smooth peanut butter since it will keep the cookie at a constant chewiness. Personally, I use smooth peanut butter to keep the cookies at a smooth consistency.