1. Preheat oven to 325 degrees. Sift flour, cocoa powder together with baking soda, baking powder, & 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 vanilla & 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).
2. 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 or silicone baking mats on baking sheets. Roll and cut scraps once. Freeze cookies until firm, about 15 minutes. Repeat with remaining disk.
3. Bake until cookies are dry to the touch, about 12 minutes. Transfer cookies to wire racks, and let cool.
4. Sift crushed candy, and separate larger pieces from dust, reserving both. Melt chopped chocolate in a microwave proof bowl, in one minute increments, stirring in-between so as not to burn chocolate. Remove from heat. Spoon the melted white chocolate onto each cooled cookie, using the back of the spoon to smooth the chocolate to the edges. Sprinkle 1/4 teaspoon of either candy pieces (big & small) on top. Refrigerate until set, up to 3 hours. Decorated cookies are best served the same day.