Move oven racks to the middle and lower-middle positions. Preheat oven to 375 degrees F. Prepare 2 baking sheets with either parchment paper or silicone baking mats.
In a medium bowl sift together the dry ingredients: the flour, baking soda, baking powder, and salt. Set aside.
In the bowl of a stand mixer (or in a large bowl if you're using hand beaters), cream the butter, peanut butter, and sugars on medium speed for 2-3 minutes, or until nice and fluffy. Add in the vanilla and the eggs, one at a time, until well mixed.
Add the dry ingredients and mix slowly until just combined. Add the chocolate chips and peanut butter chips and mix until just combined.
Using a 3 tablespoon cookie scoop, scoop the cookies 2 inches apart onto the cookie sheets. If you want a flatter cookie, compress each dough ball lightly with your fingers.
Bake the cookies for 10-12 minutes, rotating the pans halfway through baking, or until the cookies are light golden brown but still soft in the middle.
Move cookie sheets to wire rack and cool for 30 minutes before serving.