Adjust two racks to divide the oven into thirds. Preheat the oven to 375 degrees. Have ready two ungreased baking sheets. In a small bowl, mix the sugar with the crushed candy; set aside.

In a large bowl, with an electric mixer at medium-high speed, beat together the butter, shortening, confectioners' sugar, egg, vanilla, and peppermint extract until light and fluffy, 2 or 3 minutes. With the mixer at medium-low speed, gradually add the flour, beating just until blended. Remove half of the dough from the bowl and set aside on a sheet of waxed paper. To the dough remaining in the bowl, add the red food coloring and beat until evenly colored. (At this point both of the doughs can be tightly wrapped separately in aluminum foil and refrigerated for up to a week or frozen for up to three months. If frozen, thaw in the refrigerator and bring to room temperature before proceeding.) For each candy cane, scoop 1 teaspoonful of the plain dough and the same amount of pink dough. Roll each scoop between the palms of your hands to make a 10 cm rope. Twist the ropes together and shape into a candy cane. As they are made, arrange the canes on an ungreased baking sheet, spacing them about 3 cm apart.

Bake for about 9 minutes until firm to the touch and barely golden. Reverse the baking sheets on the racks and from front to back once during baking. The moment the cookies come from the oven, sprinkle each one with the sugar-and-peppermint mixture. With a wide turner, immediately transfer the cookies to wire racks to cool completely.

Store in a tightly covered container, separating the layers with sheets of waxed paper.