Bring the cider to a boil in a small saucepan and reduce to 1/2 cup. 3 to 4 minutes. Let cool.
In a liquid measuring cup, combine the milk and vinegar. Let sit 5 minutes to curdle into buttermilk, and stir. Sift together the flour, baking powder, baking soda, cinnamon, salt and nutmeg.
In the bowl of an electric mixer fitted with the paddle attachment, cream the sugar and shortening together on high speed for 1 minute. Reduce the speed to medium and add the eggs, 1 at a time, until incorporated, then mix in the vanilla. Add the buttermilk and cooled cider and mix until combined. Add the sifted flour mixture and mix on low just until a dough comes together. Chill the dough 30 minutes in the refrigerator.
When ready to fry, heat 2 inches of oil in a deep pot to 365 degrees. Roll half the dough on a floured work surface to about 3/4 inch thickness. Cut as many doughnuts as you can with a 3 1/2 to 4 inch cutter. Use a 1 inch cutter, or slightly smaller to make the doughnut holes. You should get 6 or 7 doughnuts from each half of the dough. Chill the first half of the doughnuts in the refrigerator while you roll out the second half.
To make the glaze:
In a small bowl, whisk together the confectioners' sugar, vinegar, vanilla, and 1 tablespoon water to make a thick glaze. You should be able to drizzle the glaze from a spoon. If its too thick add a little more water.
Fry the doughnuts in 2 batches until dark golden brown, about 2 minutes per side (a little less for the doughnut holes). Drain on paper towels and bring the oil back to 365 before frying the second batch. While the doughnuts are still warm, dip the surface of one side in the glaze. Let the glaze set with the glazed side up on a cooling rack for 5 to 10 minutes. These doughnuts are best served warm or at least within 1 to 2 hours of frying.
Feel free to replace pumpkin cider which isn't always so available with apple cider.