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Generously coat the work surface with flour and turn the dough out onto the floured surface. Dust the top of the dough with more flour, then roll the dough into a 1/4-inch thick sheet. Use a 3-inch doughnut cutter to stamp out circles from the dough. Place the circle on the floured parchment-lined baking sheet. Repeat, stamping out as many circles as closely together as possible, then bring the scraps together and re-roll them into a 1/2-inch thick sheet, and stamp as many circles from the dough as possible (discard the remaining scraps); you should get about 12 donuts.
Line a rimmed baking sheet with paper towels and set a wire cooling rack on top. Pour the oil into a large saucepan set over medium-high heat; the oil should fill the pot by about 3 inches-if it doesn't, add more. Heat the oil until it reads between 345 degrees F to 350 degrees F on an instant-read thermometer. Add a few doughnuts to the hot oil, taking care not to overcrowd the saucepan (otherwise the doughnuts will stick together and not cook evenly). Cook until the doughnuts are golden brown, about 1 minute on each side, then transfer to the cooling rack to drain and cool. Repeat with the remaining dough circles, making sure the temperature of the oil returns to 345 degrees F to 350 degrees F between batches.
Whisk together the granulated sugar and remaining 2 teaspoons of cinnamon in a medium bowl. Add a cooled doughnut to the bowl and turn it to evenly coat in the cinnamon-sugar. Set it on a plate and repeat with the remaining doughnuts.