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.
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. Decorated cookies are best served the same day.