For glaze, mix confectioners' sugar and cider.
Dip doughnuts while warm;
Boil apple cider in small saucepan until it is reduced to 1/4 cup, 8 to 10 minutes; cool
Beat sugar with shortening until smooth. Add eggs and mix well, then add buttermilk and reduced cider.
Stir together flour, baking powder, baking soda, cinnamon, salt and nutmeg in another bowl. Add to liquid ingredients; mix just enough to combine. Transfer dough to lightly floured board and pat to 1/2 inch thickness. Cut with 2 1/2-3 inch doughnut cutter; reserve doiughnut holes and reroll and cut scraps.
Add enough oil or shortening to fill a deep pan 3 inches; heat to 375ºF. Fry several doughnuts at a time, turning once or twice, until browned and cooked through, about 4 minutes. Remove to paper towels with slotted spoon.