Preheat your oven to 350°F. Lightly grease two baking sheets and set aside.
In a medium bowl, cream the butter and brown sugar together until the mixture becomes light and airy (with a KitchenAid, about 3 minutes. with a manual whisk, about 5 minutes). Add in the egg and vanilla.
In another bowl, sift together the flour, cocoa powder, baking soda, and salt. Next, add the dry ingredients into the butter mixture, stirring until just combined, being careful not to overmix the dough.
Using a ½ oz cookie scoop, scoop dough onto cookie sheets, leaving about 1.5" of space in between each ball of dough. Bake for 8-10 minutes, being sure to keep an eye on the consistency of the baking cookies. Let cool completely before frosting.
In a medium bowl, whisk together the peanut butter, butter, milk, and powdered sugar. Place the frosting into a piping bag (or a ziplock bag with the corner cut off) and pipe frosting onto the underside of half of cookies, then sandwich with remaining cookies. Drizzle tops of cookies with melted peanut butter.