Line a baking sheet with parchment paper or silicone baking mats; set aside.
In a medium bowl, whisk together the flour, baking powder and salt; set aside.
Using an electric mixer, beat together the butter and sugar on medium-high speed until light and fluffy, about 2 to 3 minutes. Scrape the sides of the bowl, then add the eggs one at a time, scraping the sides of the bowl as needed. Add the cocoa powder and vanilla extract, then beat on medium speed until fully incorporated, about 1 minute, stopping to scrape the sides of the bowl once. Reduce the mixer speed to low and gradually add the flour mixture, beating only until incorporated. Fold in the chopped peanut butter cups with a rubber spatula.
Divide the dough into 12 equal portions (using a scale helps to ensure that all of the cookies will be the same size). Roll each into a ball and place on the prepared baking sheet. Pat the tops of the cookies ever so slightly (don't press down too hard), then refrigerate the baking sheet for 30 minutes.
Meanwhile, preheat oven to 375 degrees F. Bake for 18 to 24 minutes, or until the tops of the cookies feel set when touched lightly. Let cool for 5 minutes on the cookie sheet, then remove to a wire rack to finish cooling. Leftovers can be stored in an airtight container at room temperature for up to 4 days.