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Combine the water, butter, and salt in a medium-sized saucepan, heat until the butter has melted, and bring to a rolling boil.
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Remove the pan from the heat, and add the flour all at once, stirring vigorously.
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Return the pan to the burner and cook over medium heat, stirring all the while, until the mixture smooths out and follows the spoon around the pan; this should take considerably less than a minute.
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Remove the pan from the heat, and let the mixture cool for 5 to 10 minutes. It'll still feel hot, but you should be able to hold a finger in it for a few seconds. If you have an instant-read thermometer, the temperature should be below 125°F.
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Transfer the mixture to a mixer, and beat in the eggs one at a time. The batter will look curdled at first, but when you add the last egg it should become smooth. Beat for at least 2 minutes after adding the last egg.
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Pour a generous 4 cups vegetable oil (peanut oil preferred) into a 10" electric frying pan or heavy skillet set over a burner; the oil should be about 5/8" deep. If you use a smaller or larger pan, add oil to a depth of between 1/2" and 3/4". Heat the oil to 375°F.
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Scoop small (1") balls of batter into the hot oil, filling the pan but not crowding the doughnuts. A teaspoon cookie scoop, filled level, works well here.
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Fry the doughnuts for about 6 minutes. As they cook they'll turn themselves over, usually multiple times. Use a chopstick or pair of tongs to give a nudge to any that seem to be stuck on one side. After about 6 minutes, the doughnuts should be a deep golden brown.
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Transfer them from the frying pan to a paper towel-lined baking sheet to drain and cool. Repeat with the remainder of the batter.
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Use a pastry bag equipped with a plain tip to pipe jelly into the interior of the cooled doughnuts. Shake them gently in a bag with granulated or confectioners' sugar. Serve immediately; or wrap loosely and store at room temperature. The doughnuts will gradually soften as they sit.