In a bowl, mix 3 cups powdered sugar, 4 tbsp. milk, and 3/4 tsp. peppermint extract. Spread 1 heaping tsp. peppermint icing onto the flat side of 1 cookie.
Top with flat side of a second cookie to form a sandwich, pressing together to squeeze filling to the edge. Roll edge of cookie in crushed and sifted candy canes. Makes 20.