In a pot, add the apple cider and a couple cinnamon sticks and heat until boiling. Reduce down to 3/4 cup of cider.
While that is reducing, take a large mixing bowl and add the flour, brown sugar, baking powder, baking soda, salt, nutmeg, and cinnamon. Whisk together these dry ingredient until evenly distributed. Set aside.
In another smaller bowl, add the egg, apple sauce and sour cream. Whisk together until smooth.
Then add in the reduced apple cider and melted butter. Whisk again until the mixture is combined.
Now pour this mixed wet ingredients into the bowl of dry ingredients. Fold together until the wet ingredients soak into the dry and a sticky batter forms.
Cover with plastic wrap and place in the refrigerator to chill for 1 hour
After the dough has chilled, begin to heat the frying oil over medium heat. Meanwhile, roll out the chilled dough on a clean, floured surface.
Knead the dough slightly with your hands before rolling out to about 1/2 inch thick. Use the large round cookie cutter and cut out donuts. Then use a small hole, such as the round end of a large metal piping tip, to create a donut hole center.
Repeat with the rest of the dough, rolling out again any excess from cutting out rounds of dough.
In a small bowl, add the white granulated sugar and cinnamon. Whisk together until combined thoroughly.
Now place each prepared donut into the heated frying oil, placing 5 or 6 in at a time, or as many fit in the pot or pan you are using.
Fry on each side until a deep brown, a few minutes on each side. Prepare a rack with paper towel or parchment paper to drain the donuts when done.
After the donuts are done, let the excess oil drain from them for a minute. Then dip each donut into the cinnamon sugar mixture coating each fully.
Once the donut is completely coated, place onto a serving dish. Repeat with the rest of the dough until all the donuts are complete.
Serve and enjoy with apple cider or milk.