Into a bowl sift the flour, the baking powder, the salt, the cinnamon, and the nutmeg. In a small bowl whisk together the eggs, the butter, the sugar, the potatoes, the milk, the zest, and the vanilla until the mixture is combined well, add the potato mixture to the flour mixture, and stir the dough until it is just combined.
Chill the dough, covered, for 1 hour, or until it is cold and can be handled easily. Roll out half the dough 1/2-inch thick on a well-floured surface and with a 3 to 3 1/2-inch doughnut cutter cut out doughnuts, reserving the center pieces. With the other half of the dough either cut out round doughnuts in the same manner or shape the dough into crullers.
For the crullers, roll the dough into a 1/2-inch-thick rectangle about 14 by 5-inches and cut into 5 by 1/2-inch strips. To form each cruller, twist 2 strips of dough together, and pinch the ends to secure them. Fry the round doughnuts, the reserved doughnut centers, and the crullers in batches in 2 inches of 375 degrees F. oil, or until they are golden, transferring them as they are fried to paper towels to drain.
Make the coating: In a shallow bowl stir together the sugar and the cinnamon
While the doughnuts are still warm, roll them, 1 at a time, in the sugar mixture, coating them well. The doughnuts keep, wrapped in plastic wrap, for 1 day.