Place the chicken into a large soup pot, and pour in the water. Add salt and 2 tablespoons butter, bring to a boil, and reduce heat to medium-low. Simmer the chicken until the meat is very tender, 2 1/2 to 3 hours. Remove the chicken from the stock, and allow the chicken and stock to cool. Separate the chicken meat from the skin and bones, and chop the meat into bite-size pieces. Strain chicken stock through a wire mesh strainer into a large pot; set aside.
Bring all but 2/3 cup of the reserved chicken stock to a boil over medium heat. In a separate saucepan, bring 2/3 cup of chicken stock to a boil. Stir in 1/2 teaspoon salt and 4 teaspoons of butter, and bring the mixture to a boil over medium heat. Beat in the flour to form a slightly sticky dough.
Remove dough to a well-floured surface, and roll to desired thickness. Knead in more flour if dough is too sticky. Cut the dough into strips about 4 inches long; drop dumplings into the boiling chicken stock. Reduce heat to low, and simmer the dumplings until tender, about 10 minutes. Stir 1 more tablespoon of butter into the stock to add richness, if desired; return chopped chicken meat to the pan. Stir in evaporated milk; simmer until chicken meat is hot.
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