Beat the cookie butter and granulated sugar until smooth, then beat in egg and vanilla. Whisk flour, cocoa, baking soda and salt and mix into the dough.
Roll the dough between parchment to 1/8 inch thick and refrigerate flat for 30 minutes, additional to active prep time. Heat the oven to 350°F and line two sheets with parchment.
Cut 36 rounds with a 2-inch cutter, rerolling scraps once. Bake one sheet at a time for 9 to 10 minutes, until the edges are firm, and cool completely.
Beat the filling butter, powdered sugar, milk and peppermint extract until fluffy. Fold in the crushed candy.
Spread the filling evenly over the flat sides of 18 cookies and sandwich with the remaining rounds. Let stand for 30 minutes so the filling firms before stacking.