Place a rack in the middle third of the oven and preheat to 350 F. Line several large baking sheets with parchment paper.
Place the unwrapped candies in a triple thickness of plastic baggies. Close the bags tightly. Using a kitchen mallet, rolling pin, or back of a heavy spoon, pound the candy into scant 1/4 inch or finer pieces. Prepare enough to fill 1/2 cup; set aside. By hand or in a processor, chop the chocolate or chocolate morsels into 1/8-inch or finer bits; set aside.
In a large mixer bowl with the mixer on medium speed, beat together the butter, sugar, and half the crushed candy pieces until well blended and fluffy, about 2 minutes. Add the egg and yolk, cocoa powder, baking powder, salt, peppermint or almond extract (if using), and vanilla and chopped chocolate and continue beating on low, then medium speed until very well blended, about 1 minute longer. Working on low speed, beat in half the flour until incorporated. Stir in the remaining flour until thoroughly incorporated. If the dough seems too sticky to handle, let stand to firm up about 5 minutes.
Divide the dough into thirds; on wax paper shape each portion into a 6-inch, evenly-thick log. Cut each log into 12 equal 1/2-inch portions. With lightly greased hands, shape the portions into balls, spacing about 2 inches apart on baking sheets. With the heel of the hand, pat down the balls until just flattened.
Bake, one sheet at a time, for 8 to 11 minutes, just until the cookies are beginning to feel firm when pressed at the edges; be very careful not to overbake. Sprinkle each cookie with a little of the reserved candy bits. Return the pan to the oven for about 40 to 50 seconds, or until the candy bits just begin to melt. Transfer the pan to a wire rack and let stand until the cookies are cooled completely. These are best fresh, but will keep, airtight and at room temperature for up to 3 days. Or freeze, airtight, for longer storage.