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1
Bring 1 cup apple cider to a boil in a small saucepan; boil until reduced to 1/4 cup, 20 to 30 minutes. Cool.
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2
Beat butter in a bowl with an electric mixer until creamy, 1 to 2 minutes. Add 1 cup white sugar and beat until light and fluffy, 2 to 3 minutes.
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3
Beat eggs, 1 at a time, into butter mixture until completely incorporated. Mix buttermilk and reduced cider into butter mixture.
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4
Whisk 3 1/2 cups flour, baking powder, baking soda, 1/2 teaspoon cinnamon, salt, and nutmeg together in bowl. Add flour mixture to buttermilk mixture; stir until dough is just combined.
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5
Transfer dough to a lightly floured piece of parchment or wax paper. Sprinkle dough with flour. Turn dough over onto a baking sheet and sprinkle with flour.
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6
Roll dough to 1/2-inch thickness, sprinkling with more flour as necessary. Transfer dough on the baking sheet to the freezer until slightly hardened, about 20 minutes.
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7
Cut cold dough using a 3-inch doughnut cutter. Gather, roll, and cut any dough scraps. Transfer cut doughnuts and doughnut holes to a second, slightly floured, baking sheet. Refrigerate doughnuts for 20 to 30 minutes.
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8
Whisk 1 cup white sugar and 1/2 teaspoon cinnamon together in a bowl. Transfer cinnamon sugar to a paper bag. Whisk 1 cup confectioners' sugar and 2 tablespoons apple cider together in another bowl. Whisk 1 1/4 cups confectioners' sugar, maple syrup, and vanilla extract together in a third bowl.
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9
Heat oil in a deep-fryer or large saucepan to 350 degrees F (175 degrees C).
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10
Cook doughnuts and doughnut holes in batches in preheated oil, turning once, until golden brown and cooked through, about 1 minute per side. Set cooked doughnuts on paper towels to drain.
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11
Shake warm doughnuts in the paper bag with cinnamon sugar, or pour cider glaze or maple syrup glaze over warm doughnuts.
