Pour the 1 cup of apple cider into a small saucepan. Set over medium heat and bring to a boil. Lower the heat to a simmer and allow the mixture to reduce to ½ cup. The cooking time will vary from 10 to 15 minutes. Remove from heat and allow the concentrated apple cider to cool.
In a small bowl, whisk together the cooled apple cider, buttermilk, and egg. Set aside.
In a large bowl, whisk together the flour, cornstarch, baking powder, baking soda, salt, cinnamon, nutmeg and sugar. Add half of the liquid mixture and stir to combine. Add the remaining liquid and stir until the dough starts to form. Add in the shortening (or softened butter) and mix until well incorporated. The dough will be very sticky. Do not add additional flour.
Cover the dough with plastic wrap, allowing the plastic wrap to rest directly on top of the dough. Chill in the fridge for 1 hour, and up to 24 hours.
Attach a candy thermometer to a large heavy-bottomed pot (such as cast-iron or Dutch oven). Fill the pot about 3 inches deep with vegetable shortening or oil. Set over medium heat. Bring the frying liquid to a temperature of 360 degrees F. Cut the doughnut holes while the oil heats.
Transfer the chilled dough to a floured work surface. Lightly flour the top of the dough and pat or roll the dough into a thickness of ½-inch. Using a floured 1½-inch round cutter, stamp out the doughnut holes. Transfer to a baking sheet. Gather the scrap dough, knead together, and re-roll to stamp out additional holes.
Brush off any excess flour from the doughnut holes. Carefully place 4 to 5 doughnut holes into the hot oil. Do not overcrowd the pot. Cook for 60 to 75 seconds on each side, until golden brown in color. Drain the cooked doughnuts on a wire rack or paper towel-lined baking sheet. Keep an eye on the candy thermometer and adjust the heat accordingly to maintain a constant cooking temperature.
While the doughnut holes are still warm, toss in a cinnamon sugar mixture to create a coating on each doughnut hole. These are best enjoyed the same day they are cooked.
Notes
It's best to use a candy thermometer to maintain proper cooking temperatures.
Frying in vegetable shortening will result with a crispier doughnut. They will also feel less greasy than oil-fried doughnuts.
Sacrifice a few doughnut holes to determine the exact length of time the dough needs to cook. Use a timer for keep consistent.