Step 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.
Step 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.
Step 3
Beat eggs 1 at a time into butter mixture until completely incorporated. Mix buttermilk and reduced cider into butter mixture.
Step 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.
Step 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.
Step 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.
Step 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.
Step 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.
Step 9
Heat oil in a deep-fryer or large saucepan to 350 degrees F (175 degrees C).
Step 9
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.