Whisk the flour, baking powder, baking soda, and salt in a medium bowl, then set the mixture aside. Blend the leaveners evenly through the flour so the cookies rise consistently.
Beat the softened butter with a hand mixer or stand mixer fitted with the paddle attachment on medium-high speed for about 1 minute. Stop when the butter looks smooth and creamy with no firm lumps.
Add the granulated sugar and brown sugar, then beat on medium-high for about 2 minutes until fluffy. The mixture should look lighter in color and feel aerated, which helps create tender edges.
Beat in the eggs, creamy peanut butter, and vanilla on high speed for about 1 minute, scraping the bowl as needed. Keep mixing until the peanut butter is fully blended and the batter looks smooth and cohesive.
Mix in the dry ingredients on low speed, then add the oats and chocolate chips while the mixer runs on low. Stop as soon as no dry flour remains because overmixing can make the cookies tough; the dough will be thick and sticky.
Cover the dough and refrigerate it for at least 20 minutes, then heat the oven to 350°F (177°C) and line baking sheets with parchment paper or silicone baking mats. After chilling, the dough should feel firm but slightly tacky rather than loose or greasy.
Scoop 2 Tablespoons of dough per cookie, place the mounds 3 inches apart, and bake for 12 to 14 minutes before cooling them on the baking sheet for 5 minutes. Pull the cookies when the sides are lightly browned and smell nutty and caramelized, while the centers still look very soft.