Combine apple cider and grated apple in a small saucepan over medium-high, and cook, stirring occasionally, until reduced to about 1/2 cup, about 25 minutes. Transfer to a bowl, and cool to room temperature. Stir in apple butter, vinegar, and vanilla.
Beat butter and sugar with an electric stand mixer on medium until light and fluffy, about 1 minute. Add egg, and beat until combined. Whisk together flour, baking powder, cinnamon, and salt; add to butter mixture alternately with apple cider mixture, beginning and ending with flour mixture. Beat on low just until blended after each addition. Turn dough out onto a floured surface, and roll into a 12- x 9-inch rectangle, about 1/2 inch thick.
Pour 2 inches of oil into a large Dutch oven, and heat over medium-high to 350°F.
Cut dough using a 2 1/2-inch round cutter. Use a 1-inch round cutter to cut out the middle of dough rounds, reserving the cutouts as doughnut holes. Reroll scraps once. Fry in batches until golden brown, about 1 minute per side for doughnuts and 45 seconds per side for holes. Transfer with a slotted spoon to a paper towel-lined baking sheet. Toss warm doughnuts in Pumpkin-Cinnamon Spice. Or cool to room temperature and dip in Buttermilk Glaze or Maple-Vanilla Icing.
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