Line a baking sheet with parchment paper. Set aside.
To the bowl of a stand mixer fitted with the paddle attachment, add 9 cups confectioners' sugar, corn syrup, water, peppermint extract, butter, and salt. Mix at medium speed until just combined.
Dust a work surface with the remaining cup of confectioners' sugar. Place the mixture onto the sugar, kneading until smooth.
Set out a sheet of parchment paper. Place patty mixture on parchment paper and sprinkle the top with more confectioners' sugar.
Place a second sheet of parchment paper on top of the filling mixture. Roll the filling out to a ½-inch thickness.
Using a 2-inch round cutter, cut out as many circles as possible and transfer to the lined baking sheet. Then, combine the leftover scraps and roll back out again. Repeat until all of the patties have been cut. (I was able to make 40 patties.)
Place in the freezer for about 15 minutes, or until firm. Near the end of the chilling time, prepare the chocolate coating.