In a small saucepan over medium heat, gently reduce apple cider to about 1/4 cup, 20 to 30 minutes. Set aside to cool.
Meanwhile, combine the flour, baking powder, baking soda, 1/4 teaspoon cinnamon, salt, and nutmeg in a bowl; set aside.
Using an electric mixer fitted with a paddle attachment, beat butter and 1/2 cup granulated sugar until smooth. Add the egg and continue to beat until the egg is completely incorporated. Use a spatula to occasionally scrape down sides of bowl. Reduce speed to low and gradually add 2 tablespoons reduced apple cider and the buttermilk, mixing just until combined. Add the flour mixture and continue to mix just until dough comes together.
Sprinkle a piece of parchment generously with flour. Turn dough onto sheet and sprinkle the top with flour. Flatten dough with your hands until it is about 1/2 inch thick. (Use more flour if the dough is still wet.) Place on a baking sheet and wrap tightly in plastic. Refrigerate overnight.
Roll dough to 1/4 inch thick. Using a 1 1/2-inch round cutter, cut out doughnuts. Using the 1/2-inch round end of a pastry tip, cut out centers. Place on a parchment-lined and flour-dusted baking sheet. Refrigerate doughnuts for 20 to 30 minutes.
While doughnuts are resting, whisk together the remaining reduced cider and the confectioners' sugar in a small bowl to make the glaze. Stir together 2 tablespoons granulated sugar and 1/4 teaspoon cinnamon in a small bowl for dusting.
Fill a deep-sided pan with 3 inches of oil. Heat over medium heat until oil reaches 360 degrees on a candy thermometer.
Line a baking sheet with several layers of paper towels. Carefully add a few doughnuts to the oil, being careful not to crowd the pan, and fry until golden brown, about 60 seconds. Turn doughnuts over and fry until the other side is golden, 30 to 60 seconds. Drain on paper towels. Dip the tops of the warm doughnuts into the glaze, or coat entire doughnuts with cinnamon-sugar mixture. Serve immediately.