To Make the doughnuts: In a saucepan over medium heat, reduce the apple cider to about 1/4 cup, 20 to 30 minutes, stirring occasionally. Set aside.
Meanwhile, in a bowl, combine the flour, baking powder and soda, cinnamon, salt and nutmeg. Set aside.
Using an electric mixer on medium speed (with the paddle attachment, if using a standing mixer) beat the butter and granulated sugar until the mixture is smooth. Add the eggs, one at a time, and continue to beat until the eggs are completely incorporated. Use a spatula to scrape down the sides of the bowl occasionally. Reduce the speed to low and gradually add the reduced apple cider and the buttermilk, mixing just until combined. Add the flour mixture and continue to mix just until the dough comes together.
Line two baking sheets with parchment or wax paper and sprinkle them generously with flour. Turn the dough onto one of the sheets and sprinkle the top with flour. Flatten the dough with your hands until it is about 1/2 inch thick using more flour if the dough is still wet. Transfer the dough to the freezer until it is slightly hardened, approximately 20 minutes. Remove the dough from the freezer. Using a 3-inch or 3 1/2-inch doughnut cutter — or a 3 1/2-inch round cutter for the outer shape and a 1-inch round cutter for the hole cut out doughnut shapes. Place the cut doughnuts and doughnut holes onto the second sheet pan. Refrigerate the doughnuts for 20 to 30 minutes.
Add enough oil or shortening to a deep-sided pan to measure a depth of about 3 inches. Attach a candy thermometer to the side of the pan and heat over medium heat until the oil reaches 350°F*. Line a plate with several layers of paper towels.
While the cut doughnut shapes are in the refrigerator, make the glaze by whisking together the confectioners’ sugar and the cider until the mixture is smooth; make the cinnamon sugar by mixing the two together. Set aside.
Carefully add a few doughnuts to the oil, being careful not to crowd the pan, and fry until golden brown, about 60 seconds. Turn the doughnuts over and fry until the other side is golden, 30 to 60 seconds. Drain on paper towels for a few minutes after the doughnuts are fried and dip the top of the warm doughnuts into the glaze or cinnamon sugar mixture. Serve immediately.