Toss the flour, cocoa powder, baking soda, and salt together in a large bowl. Set aside.

Using a handheld or stand mixer with a paddle attachment, beat the butter on high speed until creamy, about 1 minute. Scrape down the sides and the bottom of the bowl as needed. Switch the mixer to medium speed and beat in the brown sugar, granulated sugar, and peanut butter until combined. Beat in the egg and vanilla extract, scraping down the sides and bottom of the bowl as needed.

Turn the mixer off and pour the dry ingredients into the wet ingredients. Turn the mixer on low and slowly beat until a soft dough is formed. Beat in the milk on medium speed. Once mixed, beat in the peanut butter cups until evenly disbursed. The dough will be very thick and sticky. Cover the dough tightly with aluminum foil or plastic wrap and chill for at least 1 hour.

Preheat oven to 350°F (177°C). Line two large baking sheets with parchment paper or silicone baking mats. Set aside.

Drop 8 rounded balls of dough, about 1.5 Tablespoons of dough each (I like to use a medium cookie scoop), onto prepared baking sheets. Bake each batch for 10-12 minutes. Remove from the oven. The cookies will not have spread much, so gently press down on them with the back of a fork or spoon. Allow to cool on the cookie sheet for 5 minutes before transferring to a wire rack to cool completely.

As the cookies are cooling, melt the peanut butter. Drizzle over warm or cool cookies. Allow peanut butter to set for about 10 minutes. Store cookies in an airtight container at room temperature for up to 1 week.