In a sauce pan bring the apple cider to a boil and cook until reduced to one cup. Transfer it into a bowl and add 4 tbsp of butter to it and let it melt. Let it cool to about 110F.
While the cider is cooling, in a bowl of a stand up mixer combine the warm water, yeast, and half a cup of flour and mix well. Cover the bowl and set aside for about 30 minutes.
After about 30 minutes turn the mixer on low and add the egg, cinnamon, cardamom, salt and cider to the batter. Switch to a dough hook and add rest of the flour to the mix. Knead on medium low for five minutes until the dough is forming a ball around the hook, I actually needed to add a few more tablespoons of flour, but pay attention to how your dough looks.
Kneed for one minute to create a smooth bowl adding just a bit of flour if you need to, then place into an oiled bowl and let proof until doubled in size, for about 1-1 1/2 hours.
When the dough is doubled, without kneading it roll it into 1/2" thickness and cut out your donuts. I did both the donuts and donut holes. Re-roll the dough and cut your donuts and donut holes until all the dough is used.
Cover the donuts and let them rise until almost doubled in size, about one hour.
Heat 2 quarts of oil to 350F in a large frying pot. Fry the donuts (mine is a wide pot so I did 4 at a time) about 2-3 minutes per side. As soon as they are done roll them in cinnamon sugar mixture.