Combine butter, sugar and egg in bowl; beat at medium speed until creamy. Add milk, vanilla and peppermint extract; continue beating until well mixed. Add flour and baking powder; beat at low speed until well mixed. Stir in crushed candy canes.
Divide dough in half; flatten each half slightly. Wrap each in plastic food wrap; refrigerate at least 2 hours or until firm.
Heat oven to 400ºF. Line cookie sheets with parchment paper; set aside.
Roll out dough, one-half at a time, on lightly floured surface (keeping remaining dough refrigerated) to 10x4-inch rectangle. Cut rectangle in half to make 2 (10x2-inch) rectangles. Make 1 1/4–inch wide cuts across both rectangles, creating 16 (2x1 1/4-inch) pieces. Place 1 inch apart onto prepared cookie sheets.
Bake 6-8 minutes or until edges are very lightly browned. Cool 1 minute on cookie sheets; remove to cooling rack. Cool completely. Set aside 16 cookies.
Line cookie sheets with parchment paper again. Melt candy coating as directed on package.
Working with 1 cookie at a time, spread bottom-side of cookies with melted candy coating; immediately attach 2 candy canes, curved-side down. Turn over; carefully place onto prepared cookie sheets, repositioning candy canes, if necessary, to allow sled to sit level. Let stand until set.
Variation: Sparkling Sugar Cookies. Bake cookies as directed. Spread candy coating over top of each cookie. Immediately sprinkle each with white edible glitter or white sanding sugar.