Begin by browning your butter. Melt the butter in a medium saucepan or skillet over medium heat. Once it has melted, start whisking constantly for 3-4 minutes. Watch carefully, as it can burn easily. The butter will bubble, get foamy, and develop a golden brown color with a nutty aroma. Once the brown bits start to appear, keep cooking for about a minute, until the butter is golden brown, but not too dark. Remove from heat immediately and transfer to a heat-safe bowl. Let cool for several minutes, then transfer to the freezer to chill for 30 minutes, until the butter solidifies.
Preheat the oven to 350℉. Make the cookie dough by adding the chilled butter to the bowl of a stand mixer fitted with the paddle attachment, scraping all the brown bits into the mixer. Beat for 1 minute on medium-high speed. Add the brown sugar and granulated sugar, and cream together with the butter for another couple of minutes, until combined, scraping down the bowl as necessary. Add the eggs and vanilla, beating on medium speed for another 1-2 minutes, until light and fluffy.
In a separate medium mixing bowl, whisk together the flour, cocoa powder, baking soda, and salt. Gradually add the dry ingredients to the wet, mixing on low to medium speed, until incorporated. Do not overmix. Add the chocolate chips and mix for a couple of seconds. Add the chopped peanut butter cups and mix briefly, just until mixed. Use a rubber spatula to fold the dough together if needed.
Scoop the cookie dough into 1 1/2 tablespoon sized scoops, rolling with your hands to create dough balls. Place 2 inches apart on a parchment or silpat lined baking sheet. Stud the tops of the dough balls with extra chocolate chips and peanut butter cup pieces, if desired. Bake for 10 minutes or until just set – be careful not to over-bake. They will set up as they cool. Repeat with remaining dough. Cookies will stay fresh at room temperature in a sealed container for one week.