1) Sift together the flour, baking soda, salt, baking powder, and nutmeg. Set aside.
2) In a separate bowl, cream together the shortening and sugar. Add the eggs and mashed potatoes to the shortening mix.
3) Measure out the milk. Add the flour mixture and milk alternately to the creamed mixture, beginning and ending with the flour mixture.
4) The resulting dough will be quite soft, more like a drop-biscuit dough instead of a "rollable" dough. The dough can be covered and refrigerated overnight, if desired.
5) Turn the dough out onto a very well floured surface and knead a few times to make it stiff enough to roll with a rolling pin. Roll the dough using a well-floured rolling pin until it's about 1/2" thick. Dip a doughnut cutter in flour (each time you cut), and cut out the doughnuts. Save the holes; or re-roll them with leftover dough. Try to handle the dough as little as possible; the less you work the dough, the more tender the doughnuts will be.
6) Heat the fat to 365°F. Test the temperature with a thermometer; or with a slice of bread, as follows: lower half a slice of bread into the hot fat. If it takes 1 minute for the bottom side to brown nicely, the fat is at the best temperature for frying doughnuts.
7) Fry doughnuts about 2 minutes on each side or until golden brown. Drain on brown paper bags or paper towels. Check one of the fried donuts after it is cooled to be sure the center is cooked through.
8) While still warm, shake doughnuts in cinnamon-sugar or plain granulated sugar.
Yield: About 2 dozen doughnuts (plus holes), depending on the size of the cutter.