In a mixing bowl, combine the confectioners’ sugar, butter, 2 teaspoons of the peppermint extract, and the half and half and mix on medium speed until a creamy dough forms. Flatten the dough into a disk and wrap it in plastic wrap. Place in the freezer for 30 minutes.
Meanwhile, line a small baking sheet with parchment. Add the chocolate and the remaining teaspoon of peppermint extract to a small saucepan and heat on low, stirring occasionally to be sure the chocolate does not burn. When the chocolate is fully melted, remove the pan from heat and set aside.
Sprinkle a clean work surface with additional confectioners’ sugar. Roll out the dough until it is about 1/4-inch thick. Using a 1-inch cookie cutter, cut out as many candy pieces as possible. If the dough becomes sticky, dust it with additional confectioners’ sugar or place it back in the freezer to firm up.
Dip the patties into the warm chocolate and carefully remove each one using a fork. Place the chocolate covered patties onto the parchment lined baking sheet. While the chocolate is still warm, sprinkle the patties with the crushed candy canes. When all candies are decorated, place the baking sheet into the freezer for at least 20 minutes before serving.