In a small saucepan, boil apple cider over medium-high heat until reduced to ½ cup, which will take about 30 minutes. Allow to cool completely and then whisk together with apple butter, buttermilk, and vanilla. Set aside.
In a medium bowl, whisk together all dry ingredients (flour, baking powder, baking soda, salt, cinnamon, nutmeg, and allspice). Set aside.
In a large bowl, whisk together eggs and both sugars until lighter in color, 1-2 minutes. Slowly stream in melted butter to emulsify.
Add ⅓ of the dry ingredients, followed by half of the wet ingredients. Continue alternating dry/wet, beginning and ending with dry. You may need to switch to a spatula at the end because the batter/dough will be very thick.
Sprinkle flour onto a sheet of parchment paper and dump dough out onto the parchment. Heavily sprinkle the top with more flour, cover with another sheet of parchment, and gently press or roll the dough into a flatter rectangle (it is very soft, so you may need to just gently push it along with your rolling pin versus actually rolling it out).
Slide onto a baking sheet and freeze for 10-15 minutes. Meanwhile, cut out parchment squares that are slightly larger than your donuts will be and arrange them on a baking sheet. In a shallow bowl, mix cinnamon & sugar together and set aside.
Roll out chilled dough, still covered in parchment, into a flatter rectangle that's roughly ¼-½ inch thick. Remove the top parchment and brush off any excess flour.
Cut out circles with a well-floured round cutter (such as a biscuit cutter) and place onto parchment squares. Take a smaller (1-inch in diameter) well-floured round cutter and cut out the middle (it's much easier to do this when the rounds are already on the squares of parchment).
Heat oil in a Dutch oven or deep fryer to 330° F. Fry donuts for 2-3 minutes per side, or until deep golden brown. Remove and place on wire rack or paper towel-lined plate briefly to drain before dredging in cinnamon/sugar mixture while still warm.
Notes
*Donuts may be frozen raw by placing on parchment-lined baking sheet and flash freezing for about an hour. Remove and place in large ziptop bag. Freeze for up to 2 months. Thaw, covered, before frying and dredging in cinnamon/sugar. **Freezing fried donuts is done the same way, without dredging in cinnamon sugar mixture. Thaw, covered, and then microwave for a few seconds (10-20) before dredging in cinnamon sugar. ***Donuts are always best when served within a few hours of frying, but they can be reheated. To maintain the crisp crust, air fry at 350 degrees F for 3-4 minutes. ****I know many of you out there will ask me about baking these donuts. While I don't think there is such a thing as a baked donut (donuts are meant to be fried), you can try piping the thick batter into a greased donut pan and baking the donuts at 350 degrees F for 10-12 minutes. Brush or dip the donuts in melted butter before dredging in the cinnamon sugar coating.