The night before, soak beans in cold water to cover by 2 inches. Change water at least once. Drain and rinse.
In a heavy kettle, cook onion, celery, parsley, thyme, and bay leaf in the butter over mod-low heat, stirring, for 10 minutes. Add ham hock, beans, broth, 4 cups water, salt and pepper to taste. Bring the mixture to a boil, reduce heat, and simmer, uncovered, adding more water if necessary to keep beans covered, for 3 hours.
Discard ham hock and bay leaf. Put the mixture through a medium disk of a food mill into a large bowl and then return it to the kettle.
Stir in rum, lemon juice, and salt and pepper to taste. Adjust the consistency with hot water and garnish for serving with the eggs, parsley, and lemon slices. Makes about 7 cups.