Preheat oven to 325 degrees. Sift flour, cocoa powder, baking soda, baking powder, and salt into a bowl.
Beat butter and sugar with a mixer on medium-high speed for 1 minute. Reduce speed to medium-low, and add egg, then yolk, beating well after each addition. Beat in peppermint extract.
Slowly add flour mixture, and beat until just incorporated. Shape dough into 2 disks, wrap each in plastic, and refrigerate until firm, at least 1 hour (or up to 2 days).
Roll out 1 disk of dough on a lightly floured surface to 1/8-inch thickness. Freeze until firm, about 15 minutes.
Using a 2-inch round cutter, cut out circles, and place 1 inch apart on parchment-lined baking sheets. Roll and cut scraps once. Freeze cookies until firm, about 15 minutes. Repeat with remaining disk. Alternately, form dough into tablespoon-sized balls with your hands. Freeze the balls for about 15 minutes, then press flat with a glass.
Bake until cookies are dry to the touch, about 12 minutes. Transfer parchment, with cookies, to wire racks, and let cool. (Undecorated cookies will keep, covered, for up to 3 days.)
Sift crushed candy, and separate larger pieces from dust, reserving both.
Melt chocolate in a heatproof bowl set over a pot of gently simmering water. Remove from heat. Dunk cookies into melted chocolate. Using a fork, turn to coat, let excess drip off, and gently scrape bottom against edge of bowl. Place on parchment-lined baking sheets, and sprinkle 1/4 teaspoon of either candy pieces or dust on top. Repeat, sprinkling half the cookies with pieces and the rest with dust. Refrigerate until set, up to 3 hours.
Note: Andes Peppermint bark works great in place of crushed candy canes.