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Sprinkle the rice and salt into the boiling water.
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Cover and cook slowly until the water is absorbed, 7-10 minutes.
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Add the milk, stir, cover, and cook 10-12 minutes more, or until the rice is tender.
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Stir in 1 egg and 2 tablespoons of the butter.
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Spread the mixture on a shallow plate, cover with plastic wrap, and refrigerate.
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Beat the remaining egg in a shallow dish and put the crumbs on a piece of wax paper.
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Shape the well-chilled rice mixture into 6 conical or patty shapes, pressing firmly together.
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Carefully dip each cake into the egg and then cover with crumbs.
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Melt the remaining 4 tablespoons of butter in a skillet with the oil.
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When hot, fry the cakes until golden brown; don't cook them too quickly or the insides will remain cold.