Whisk the flour, cocoa powder, baking powder and salt in a small bowl to blend and set aside.
In the bowl of an electric mixer, combine the butter and sugar. Beat on medium-high speed until light and fluffy, 2-3 minutes. Add the egg, peppermint extract, and vanilla and blend well. With the mixer on low speed, add the dry ingredients and mix only until incorporated and no streaks remain. Scrape dough out onto plastic wrap and form the dough into a disc, wrap tightly with the plastic wrap and refrigerate until firm, 1-2 hours or overnight.
Preheat the oven to 325˚ F. Line a baking sheet with parchment paper or a silicone baking mat. On a lightly floured work surface (don't worry flour will come off), roll the dough out to a little less than ¼-inch thick. Cut out 2-inch rounds with a cookie cutter and place on the prepared baking sheet. Bake 10-12 minutes, until set. Transfer to a wire rack to cool completely.
For the Chocolate Coating
Line baking sheets with wax or parchment paper. Place the chocolate in a heatproof bowl set over an inch or two of simmering water- the bottom of the bowl should not touch the water.
Melt chocolate, stirring occasionally until smooth or you can temper the chocolate so it sets better.
Dip the bottom side of each cookie into the chocolate. Use an offset spatula to smooth off the excess. Place chocolate side up on the prepared baking sheets. Place in the refrigerator to chill until the chocolate has set. Repeat process with the top of the cookie. OR Alternately- to coat entire cookie at once- place the cookie right in the chocolate and make sure it is fully coated by pressing it into melted chocolate with a fork. Lift it out of the chocolate by resting it on the fork, tapping the fork on the side of the bowl to get off excess. Use the offset spatula to get excess off top of cookie and then gently wipe bottom of fork on the side of bowl. Place carefully on prepared pan.
If desired, sprinkle crushed candy candy cane on top of cookies before chocolate is set.
Notes
If you really don't like rolling out cookie dough- you can chill the dough till a little firm- roll it into 2 logs the size you want your cookies and wrap in plastic wrap. Then refrigerate until really firm- even overnight. Slice into between 1/8 and 1/4 inch circles with a sharp knife. You may have to gently press them into a better circle shape with the roller after cut. Cookie base adapted from Annies Eats originally from Martha Stewart