FOR THE COOKIES: Adjust oven racks to upper-middle and lower-middle positions and heat oven to 350 degrees. Line 2 baking sheets with parchment paper. Pulse peanuts in food processor until finely chopped, about 8 pulses. Whisk flour, baking soda, and salt together in bowl. Whisk butter, peanut butter, granulated sugar, brown sugar, milk, and egg together in second bowl. Stir flour mixture into peanut butter mixture with rubber spatula until combined. Stir in peanuts until evenly distributed.
Using #60 scoop or tablespoon measure, place 12 mounds, evenly spaced, on each prepared baking sheet. Using damp hand, flatten mounds until 2 inches in diameter.
Bake until deep golden brown and firm to touch, 15 to 18 minutes, switching and rotating sheets halfway through baking. Let cookies cool on sheets for 5 minutes. Transfer cookies to wire rack and let cool completely, about 30 minutes. Repeat portioning and baking remaining dough.
FOR THE FILLING: Microwave peanut butter and butter until butter is melted and warm, about 40 seconds. Using rubber spatula, stir in confectioners’ sugar until combined.
TO ASSEMBLE: Place 24 cookies upside down on work surface. Place 1 level tablespoon (or #60 scoowarm filling in center of each cookie. Place second cookie on top of filling, right side up, pressing gently until filling spreads to edges. Allow filling to set for 1 hour before serving. Assembled cookies can be stored in airtight container for up to 3 days.