Add the apple cider and cinnamon stick to a saucepan on the stove, and heat on medium.
Simmer the cider until it's down to about 2/3 of a cup. White the cider is simmering, you can start prepping the rest of the ingredients. Note: if you reduce it too far, top it off with a little fresh cider.
Preheat your oven to 350℉. Spray donut pans with cooking/baking spray.
Whisk together all the dry ingredients (flour, salt, baking powder, baking soda, and spices).
In the bowl of a stand mixer, whisk together the brown sugar, sugar, and oil on medium/low. Add the eggs one at a time, fully combining after each addition, then the vanilla.
Once the cider is cool enough to touch, add it to the sugar/egg mixture.
Add half of the dry ingredients to the wet, then stir in the milk. Add the rest of the dry ingredients, then finish by adding the applesauce last.
Into a gallon Ziploc bag or large pastry bag, pour the batter and squeeze to the bottom. Snip the tip off, leaving a hole about ¾ of an inch wide.
Pipe the donut batter into the rings of your prepared donut pans. It's okay if they're not totally perfect - donuts are very forgiving!
Bake for 10 minutes, or until they're nicely risen and a toothpick inserted comes out dry.
Let the donuts cool for 10 minutes in the pan, then turn them out on a cooling rack. Prepare the coating.
Melt the butter in a bowl wider than the donuts. In a separate bowl of the same size, whisk together the sugar and cinnamon.
Toss the donuts in the butter, then pick them up and let as much butter as possible drip off. Once the butter is mostly off, dredge lightly in the cinnamon sugar. Store in an airtight container for up to 1 week.