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1. Place the chicken breast, thigh, and liver, the pork, and the cold water in a heavy, nonreactive 4-gallon kettle; set over moderately high heat and bring to a boil. Adjust the heat so the water bubbles gently, cover, and simmer for 30 minutes. Remove and refrigerate all pieces of chicken. Cover the kettle again and simmer the pork about 1 1/2 hours longer or until very tender.
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2. Remove the skin and bones from the chicken, cut the meat into 1-inch chunks, then pulse quickly in a food processor until the texture of coarsely ground meat (in the old days, the chicken was fed through a meat grinder). When the pork is tender, cut into 1-inch chunks, then pulse just as you did the chicken. Also pulse the chicken liver.
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3. Return the chicken, liver, and pork to the kettle, add the beans and their soaking water, the onions, tomatoes and their liquid, the corn, and peas. Cover and simmer for 1 hour.
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4. Add the salt, pepper, and butter, reduce the heat to its lowest point, and simmer the burgoo uncovered for 3 1/2 to 4 hours, stirring occasionally, or until as thick as chili.
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5. Taste the burgoo for salt and pepper, adjust as needed, then ladle into heated soup bowls.
