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Place the eggs gently in a pot or saucepan and cover them with cold water.
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Bring the water to a hard boil and boil for two minutes.
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After two minutes at full boil, cover the pot, remove it from the heat and let it stand for fifteen minutes.
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Cool the eggs under cold running water, and remove their shells.
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Combine the brown sugar, soy sauce, chicken broth and sesame oil in a saucepan.
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Heat the mixture, stirring to dissolve the brown sugar.
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Add the boiled eggs and simmer, covered, for one hour (the liquid should cover the eggs, but if it does not, baste them frequently).
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Turn off the heat and let the eggs stand in the sauce for an additional hour, turning them from time to time to ensure even coloring.
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Serve cut into halves or quarters, with the dipping sauce.