Butter a rimmed 15½x10 ½-inch baking pan (try a jellyroll pan) and set aside. (A 9x13-inch pan will also do, but the needhams will be thicker.) In a bowl or the bowl of a stand mixer, add confectioners’ sugar, potatoes, melted butter, coconut, vanilla, and salt. Mix on low speed (mixture will be very dry at first) until a thick paste forms; then increase the speed for 30 seconds, or until all of the sugar is completely incorporated.
Transfer mixture to prepared pan, and press down evenly. Let mixture set and firm up at room temperature, about 1 hour. Then cut into small, even squares.
Melt unsweetened chocolate and chocolate chips in a double boiler over not-quite-simmering water until smooth; then remove from the heat and add shortening, stirring vigorously until shiny.
Place one coconut/potato square onto the tines of a fork and lower into the chocolate until coated; lift it and let the excess chocolate drip back into the pot. Transfer square to a sheet of parchment to harden, about 1 hour. Repeat with remaining squares. Store in an airtight container for up to two days for best flavor and appearance.