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Line 2 baking sheets with parchment paper and lightly dust the sheets with flour. Line 2 additional baking sheets with paper towels.
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In the bowl of a standing mixer or in a large bowl, stir together warm milk, yeast, and 1 cup flour. Cover and set aside for 30 minutes. With a dough hook fitted to the mixer, and the motor on LOW, add eggs, evoo, vanilla extract, sugar, salt, and remaining flour. Alternatively, you can mix in ingredients by hand.
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Knead by mixer or by hand until it’s smooth and elastic. If making doughnuts the same day, place dough in a lightly greased bowl and let rise for 1 to 2 hours until doubled in size. Or, lightly oil dough, place in a resealable bag, and store in the refrigerator for up to 1 day.
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When ready to make the doughnuts, lightly flour your work surface and roll dough to ¼-inch thick. Cut dough into desired size doughnut with cookie or doughnut cutters. For a filled doughnut, do not cut out a center. If using the make ahead dough method, let dough come to room temperature before rolling and cutting.
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Place cut dough on prepared baking sheet lined with parchment paper and flour. Allow doughnuts to rise at room temperature for 1 hour.
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Pour 2 quarts of oil in a large heavy bottom pot or Dutch oven fitted with a deep fry thermometer and place on medium-high heat. Once oil reaches 360°F on the deep fry thermometer, place several doughnuts in oil and fry 2 minutes per side. Use chopsticks to gently flip the doughnuts over and fry for 1 to 2 minutes on the other side. Transfer fried doughnuts to paper towel lined sheets.