Special equipment: a 3-by-1-inch biscuit cutter or a 3-inch donut cutter
Put the apple cider and brown sugar in a small saucepan. Bring to a boil over high heat, then lower the heat to medium and cook until reduced to 1/3 cup, about 25 minutes. Transfer to a bowl and refrigerate until completely cool, about 15 minutes.
Stir together the flour, salt, baking powder, baking soda, 1 1/4 teaspoons cinnamon and the nutmeg in a medium bowl.
In a stand mixer fitted with the whisk attachment, whip the butter and 1 cup of the sugar on medium-high speed until light and fluffy, about 2 minutes. Lower the speed to medium and beat in the eggs, one at a time. Add the buttermilk, vanilla extract and the cooled apple syrup, and mix until combined. Lower the speed again and add the flour mixture; mix until just combined; the dough will be quite soft.
Line 2 baking sheets with parchment, and sprinkle with a little flour. Turn the dough out onto one of the baking sheets; gently pat out to an even 3/4-inch thickness and sprinkle the top with flour. Cover with plastic wrap and refrigerate until firm, at least 2 hours and up to overnight.
With a lightly floured 3-inch donut cutter, cut out about 18 doughnuts with holes. Transfer the donuts to the second baking sheet, and refrigerate for about 30 minutes. It is important that the dough is firm and cool before frying.
Line a large plate or another baking sheet with several layers of paper towels; set aside. Put the confectioners' sugar in a medium bowl or baking dish. In another bowl or baking dish, mix together the remaining 1 1/2 cups sugar and 2 tablespoons cinnamon; set aside.
Heat 3 inches of canola oil in a Dutch oven until it reaches 360 degrees F. In batches of 3 or 4, fry the donuts until browned on one side, about 2 minutes; flip, and cook until browned on the other side, 1 to 2 minutes longer. With a slotted spoon, transfer the donuts to the paper towels, just to blot the excess oil; then, immediately drop the donuts in either the cinnamon sugar or the confectioners' sugar, turning to coat. Transfer the finished donuts to a serving plate. When all the donuts are cooked and sugared, repeat with the donut holes. If the raw donuts get soft while you are frying, pop them in the freezer until you are ready to cook them. Serve warm.
This recipe has been updated and may differ from what was originally published or broadcast.