Bring apple cider to a boil in a saucepan over medium-high heat. Boil until reduced to about ⅓ cup. Transfer to a heat-proof bowl if desired. Stir in the apple butter. Set aside.
Whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, salt, and allspice. Set aside.
Whisk together the eggs, buttermilk, and the apple cider/apple butter mixture. Set aside.
Beat the butter, brown sugar, and granulated sugar with a stand mixer.
Stir in ½ of the flour mixture. Add ½ of the apple cider mixture and stir to combine. Repeat with the remaining dry and wet ingredients. (See notes for how to make baked donuts at this point.)
Turn onto a well-floured, parchment-covered baking sheet. Sprinkle additional flour on top of the dough. Flatten or roll out to about ¾ inch thickness. Cover and refrigerate for about 1 hour (or overnight).
Heat oil in a deep fryer (or use a large pot with a thermometer) to 360˚F.
Remove dough from the refrigerator. Use a 3 inch, floured doughnut cutter to cut out doughnuts. Save the donut holes and re-roll scraps. Return the cut donuts to the fridge until ready to fry.
Fry a couple of doughnuts at once for about 3 minutes, carefully turning halfway through cooking, until deep brown on both sides. Fry the holes for 2-3 minutes.
Drain on a paper towel lined plate. Coat with cinnamon sugar.