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 into a log the size of a paper towel holder.
Freeze log until solid, about 1 hour.
Cut log into 1/2 inch slices.
Transfer slices to a piece of wax paper, pressing down on slices to spread the patties out.
Freeze slices until firm, about 30 minutes.
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 and let set for 1 hour until chocolate firms up.