"Scrape the dough onto a lightly floured sheet of parchment paper and pat into a 7-by-11-inch rectangle, about 1/2 inch thick. Cover with plastic wrap and refrigerate for at least 2 hours or overnight.
Meanwhile, make the glaze: Simmer the remaining 1 cup cider in a small saucepan over medium heat until reduced to 1/4 cup. Whisk in the confectioners' sugar until smooth and glossy, then set aside. Mix the remaining 1 cup granulated sugar and 2 teaspoons cinnamon in a shallow bowl; set aside for the topping.
Heat 2 inches of vegetable oil in a large heavy-bottomed pot over medium-high heat until a deep-fry thermometer registers 350 degrees. Line a baking sheet with paper towels. Cut the chilled dough into 12 rounds, using a floured 2 1/2- or 3-inch biscuit cutter, then cut out the middles with a 1-inch cutter (or use a doughnut cutter). Slip 2 or 3 doughnuts at a time into the hot oil and fry until golden brown, 1 to 2 minutes per side, adjusting the heat as needed. Transfer to the paper towels to drain.
Dip one side of each doughnut in the cider glaze, letting the excess drip off; dip just the glazed side in the cinnamon-sugar or roll all over in cinnamon-sugar, if desired. Serve warm."