Bring 1 cup apple cider to a boil in a small saucepan; boil until reduced to 1/4 cup, 20 to 30 minutes. Cool.
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.
Beat eggs, 1 at a time, into butter mixture until completely incorporated. Mix buttermilk and reduced cider into butter mixture.
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.
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.
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.
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.
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.
Heat oil in a deep-fryer or large saucepan to 350 degrees F (175 degrees C).
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.
Shake warm doughnuts in the paper bag with cinnamon sugar, or pour cider glaze or maple syrup glaze over warm doughnuts.