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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 1/2-inch thick sheet. Use a 2 1/2 to 3-inch doughnut cutter to stamp out circles from the dough. Place the circles on a parchment paper-lined baking sheet. Bring the scraps together, 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 2 dozen doughnuts.
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 saucepan by about 3 inches-if it doesn't, add more. Heat the oil until it reads 375 degrees F on an instant-read thermometer. Add 2 to 3 doughnuts to the hot oil and cook until the doughnuts are golden brown, 1 1/2 to 2 1/2 minutes on each side. Transfer to the cooling rack to drain and cool and repeat with the remaining doughnuts, making sure the temperature of the oil returns to 375 degrees F between batches.
Serve the doughnuts plain or roll in cinnamon-sugar or confectioners' sugar before serving.