In a large bowl, mix together the potatoes, salt, butter, and peppermint extract. Gradually mix in confectioners' sugar; mix in enough to make a workable dough, between 6 and 8 cups.
Knead slightly, and roll into cherry-size balls. Flatten balls to form patties. Arrange on sheets of wax paper, and allow to dry overnight.
Place chocolate and shortening in a microwave-safe bowl. Heat in microwave, stirring occasionally, until melted and smooth. Dip patties in melted chocolate, and let cool on wax paper.