Pop the mini peanut butter cups into the freezer for 10 minutes to firm up for cleaner cuts. Heat the oven to 375°F (190°C) and line 2 baking sheets with parchment.
Brown the butter: Melt butter in a light-colored saucepan over medium heat, stirring, until golden with brown specks and nutty aroma (4–6 minutes). Scrape into a large bowl and cool until just warm to the touch, 5–7 minutes.
Vigorously whisk in the brown sugar and granulated sugar until glossy and slightly thick, about 60 seconds. Let the mixture rest 2 minutes, then whisk again for 20–30 seconds to help the sugars dissolve.
Whisk in the egg and vanilla until the batter looks smooth and cohesive.
In a separate bowl, whisk together the flour, cocoa powder, fine sea salt, baking soda, and espresso powder. Add this dry mix to the wet ingredients and fold with a spatula just until a few flour streaks remain.
Stir in the chopped peanuts and half of the chopped peanut butter cups, reserving the rest for topping. Chill the dough for 15 minutes to help minimize spread.
Scoop 1½-tablespoon portions of dough (medium scoop) onto the prepared sheets, spacing about 2 inches apart. Press a few of the reserved peanut butter cup pieces onto the tops of each mound.
Bake one sheet at a time for 8–10 minutes, rotating the pan halfway, until edges are set and centers look slightly soft and matte. Tap the pan once on the counter right out of the oven to settle the cookies, then finish with a pinch of flaky sea salt.
Cool on the baking sheet for 5 minutes before transferring to a rack to cool completely. Repeat with remaining dough.