Reduce Apple Cider:
In a large frying pan, bring apple cider to a boil over medium-high heat. Simmer until reduced to about 1/3 cup (around 10 minutes) and syrupy.
Remove from heat, whisk in butter and brown sugar until melted and combined. Transfer to a mixing bowl.

Combine Wet Ingredients:
Whisk applesauce, chilled buttermilk (or kefir), and cold eggs into the apple cider mixture until smooth.

Mix Dry Ingredients:
In another bowl, whisk together flour, cinnamon, baking powder, salt, baking soda, and nutmeg.
Add the wet mixture to the dry ingredients and stir gently until a sticky dough forms.
Cover and refrigerate for at least 1 hour (or overnight if using extra baking powder).

Shape Donuts:
On a generously floured surface, roll out the chilled dough to 1/2-inch thickness.
Cut donuts using a floured donut cutter (or two round cutters, about 3″ and 1″).
Place on a floured baking sheet and chill until ready to fry.

Heat Oil:
In a Dutch oven or heavy pot, heat oil to 350°F (175°C).
Maintain temperature for even frying.

Fry Donuts:
Fry 3–4 donuts at a time for about 1 minute per side, or until golden brown.
Remove with tongs or chopsticks and drain on a wire rack over paper towels.
Repeat with donut holes, frying about 1 minute per side.

Coat in Cinnamon Sugar:
While still warm, toss each donut in the cinnamon-sugar mixture until well coated.
Cool slightly on a wire rack before serving.