In a large bowl, mix together the mashed potatoes, sugar, milk, butter and eggs. Add the flour, baking powder, cinnamon, ginger and salt; stir until you have a thick, sticky dough. If you like, refrigerate your dough for an hour or so, or overnight.
Generously flour the countertop, place a piece of the dough on it, and shake flour over it before patting about 1/2-inch thick. Cut into rounds with a doughnut cutter while you heat a few inches of oil in a heavy pot until hot but not smoking – you’ll know it’s hot enough when the oil bubbles around a scrap of dough dropped in.
Cook the doughnuts a few at a time, without crowding the pot, until golden on both sides, flipping with tongs or a slotted spoon as needed. Transfer to paper towels to drain, then dust in icing sugar or roll in a shallow bowl of regular sugar to coat while still warm. Makes about 3 dozen doughnuts.