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Place the potatoes in a medium saucepan, cover with cold water by 2 inches and season generously with salt. Bring to a boil over high heat, then lower the heat to maintain a gentle simmer and cook until very tender, about 15 minutes. Reserve 1 1/2 cups of the potato cooking water. Drain and mash the potatoes until smooth. Set the mashed potatoes aside and cool slightly.
In a medium bowl, stir together the mashed potatoes, reserved potato cooking water, sugar and yeast. Cover with a tea towel and let the mixture rise for 20 minutes; it will look foamy.
Using a sieve, strain the mixture into a large bowl, smoothing out any lumps. Stir in the eggs and melted butter. Stir 5 cups of the flour, one cup at a time, until a dough starts to form. Gently knead the remaining 1 cup of flour in until the dough pulls away from the sides of the bowl and forms a large ball. Cover with a tea towel and put in a warm place to rise for 3 hours.
Dust some flour onto a baking sheet. On a well-floured surface, roll out the dough to 1/2-inch thick. Using a 3-inch cookie cutter, cut out the doughnuts and set on the prepared baking sheet. Cover lightly with a tea towel, and let rise for 30 minutes.
Line a baking sheet with a split paper bag. In a deep fryer or Dutch oven, pour enough oil to fill no more than halfway up the side. Heat the oil until a deep-frying thermometer inserted in the oil reaches 365 degrees F.
Gently place the doughnuts into the oil, two at a time. Cook until golden brown, about 2 minutes per side. Transfer the doughnuts to the prepared baking sheet to absorb some of the oil. Place the doughnuts in the remaining paper bag with a heaping spoonful of sugar and shake gently until the doughnuts are thoroughly coated. You may leave some plain to slice in half and spread with molasses if using. Serve immediately.