Peppermint Patties

Beat 2 1/4 cups confectioners sugar with corn syrup, water, peppermint extract, shortening, and a pinch of salt using an electric mixer (with paddle attachment if using a stand mixer) at medium speed until just combined. Knead on a work surface dusted with remaining 1/4 cup confectioners sugar until smooth. Roll out between sheets of parchment paper on a large baking sheet into a 7- to 8-inch round (less than 1/4 inch thick). Freeze until firm, about 15 minutes. Remove top sheet of paper and sprinkle round with confectioners sugar. Replace top sheet, then flip round over and repeat sprinkling on other side.
Cut out as many rounds as possible with cutter, transferring to a parchment-lined baking sheet. Freeze until firm, at least 10 minutes. Meanwhile, gather scraps, reroll, and freeze, then cut out more rounds, freezing them.
Melt three fourths of chocolate in a metal bowl set over a saucepan of barely simmering water. Remove bowl from pan and add remaining chocolate, stirring until smooth. Cool until thermometer inserted at least 1/2 inch into chocolate registers 80°F.
Return water in pan to a boil and remove from heat. Set bowl with cooled chocolate over pan and reheat, stirring, until thermometer registers 88 to 91°F. Remove bowl from pan.
Balance 1 peppermint round on a fork and submerge in melted chocolate, letting excess drip off and scraping back of fork against rim of bowl if necessary, then return patty to sheet (to make decorative ridges on patty, immediately set bottom of fork briefly on top of patty, then lift fork straight up). Coat remaining rounds, rewarming chocolate to 88 to 91°F as necessary. Let patties stand until chocolate is set, about 1 hour.