Bring the apple cider to a boil in a medium saucepan. Boil until it is reduced to 1⁄4 cup, 7 to 10 minutes. Pour into a glass measuring cup and let cool to room temperature. Combine the flour, yeast, cinnamon, and salt in a medium bowl and set aside.
Place the butter and sugar in the bowl of a standing mixer fitted with the paddle attachment. Cream together on medium-high speed until well-combined, about 3 minutes, scraping down the sides of the bowl once or twice as necessary. Add the egg yolks and beat, scraping down the sides of the bowl once or twice as necessary, until smooth. Add the cooled cider and the buttermilk and beat until combined. Stir in the flour mixture until just combined. Do not overmix!
Transfer the dough to a lightly oiled bowl or dough-rising container, cover the bowl with plastic wrap, and let stand at room temperature until it is puffy and slightly risen, about 1-1⁄2 hours.
Line 2 baking sheets with parchment or waxed paper and sprinkle generously with flour. Turn the dough out onto one of them and sprinkle the top of the dough with flour.
Flatten the dough with your hands until it is about 1⁄2 inch thick, sprinkling on more flour if necessary to prevent sticking. Place the baking sheet in the freezer until the dough is slightly hardened, about 20 minutes.
Remove the dough from the freezer and use a 3-inch doughnut cutter to cut as many doughnut shapes as you can. Place the cut doughnuts and the doughnut holes on the second baking sheet. Re-roll and cut the scraps. Refrigerate the cut doughnuts and holes for 20 to 30 minutes.
Add 3 inches of oil to a deep pot. Attach a candy thermometer to the side of the pot and heat the oil over medium heat until it reaches 350°F. Line a large platter or another baking sheet with several layers of paper towels.
Carefully add a few doughnuts to the pot, being careful not to crowd them. Fry until golden brown on one side, 30 seconds to 1 minute. Turn and continue to fry until golden on both sides, another 30 seconds to 1 minute. Drain the cooked doughnuts on the paper towels, and repeat with the remaining doughnuts, keeping an eye on the thermometer to make sure the oil stays at a steady 350°F and adjusting the heat as necessary.
Roll the warm doughnuts in cinnamon sugar and serve warm.
For glazed cider doughnuts, whisk together 1⁄2 cup of apple cider with 2 cups of confectioners’ sugar until smooth. Place the cooled doughnuts on a wire rack set over a rimmed baking sheet and pour the glaze over the doughnuts to coat the tops. Let them stand until the glaze has hardened, about 30 minutes, before serving.