Put the confectioners sugar in a large heatproof bowl. Make a deep well in the middle. Add the mashed potatoes, butter, vanilla, and salt to the well.
Place the bowl over a saucepan of barely simmering water. Starting in the center of the well, stir the ingredients, gradually incorporating the sugar, until the sugar dissolves and the mixture forms a smooth paste, about 5 minutes.
Remove the bowl from the heat and stir in the coconut. Transfer to the freezer and chill until firm, about 20 minutes.
Roll tablespoons of the potato mixture into balls, then flatten each ball into a small square using your fingers; you should have about 40 squares. Put the squares on a large plate or baking sheet, cover with plastic wrap and freeze until firm, about 20 minutes.
Put the chocolate in a microwave-safe bowl and microwave in 30-second intervals, stirring, until the chocolate melts.
Add the shortening and stir vigorously until glossy, about 2 minutes. Transfer half of the chocolate to a glass liquid measuring cup.
Line a baking sheet with foil, shiny-side up. One at a time, place each candy square on a fork and dip in the melted chocolate to coat; let the excess drip back into the cup. Transfer to the prepared baking sheet.
Repeat with the remaining squares, refilling the cup with melted chocolate as needed. Let the candies harden at room temperature, about 1 hour.