Preheat oven to 350 degrees F. Prepare 2 large baking sheets with parchment paper or a silicone baking mat.
In the bowl of an electric mixer fitted with the paddle attachment, or with an handheld mixer, cream peanut butter, butter, brown sugar and granulated sugar together. Add eggs, one at a time, beating well. Add baking soda, baking powder and vanilla, beating well. With the mixer on low speed, mix in flour until well blended. Gently stir in peanut butter cups.
Scoop dough into tablespoon-sized balls and place on prepared baking sheets.
Bake for 8-9 minutes, or until cookies are golden and appear slightly under-baked. Remove from oven and let cool on baking pan for 5 minutes (they will continue to bake on the hot pan, but will remain perfectly gooey and chewy!). If desired, top cookies with additional peanut butter cups.