I adore peppermint patties and never knew I could make them at home!
Beat 2 1/4 cups confectioners sugar with corn syrup, peppermint extract, shortening, and a pinch of salt using a stand mixer (with paddle attachment) at medium speed until just combined. Knead on a work surface dusted with remaining 1/4 cup confectioners sugar until smooth.
Roll mixture out between sheets of parchment paper on a large baking sheet.
Freeze until firm, about 15 minutes. Remove top sheet of parchment paper and sprinkle 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 chocolate in medium microwave safe bowl for 1 minute. Remove and stir, looking to see if all chocolate has melted. If not, return to microwave in 15 second increments until chocolate is melted.
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 cookie sheet. Coat remaining rounds, reheating chocolate if necessary.