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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 2 3/4 -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 12 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, 2 to 3 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.
For the icing: In the bowl of a stand mixer add the butter and vanilla. Using the paddle attachment, beat the butter on medium-low speed until creamy. Alternate adding 4 cups of the confectioners' sugar and the cream in three additions, starting and ending with the confectioners' sugar and adding more if the mixture doesn't form soft peaks after 10 minutes. Add the pink peppercorns to taste.