Simmer chicken breasts in water to barely cover until tender (1-½ hours, longer if necessary). Drain, saving the broth for soup. Cool chicken and remove skin and bones. Break chicken into large bite-size pieces, saving smaller pieces for soup.
Shortly before serving, heat the oil to 375 degrees F. Roll the large chicken pieces in flour. Then dip them into the milk and egg mixture. Finally roll the chicken in the cracker crumbs. Fry until golden (about 4 minutes). Since the frying time is so brief, the chicken is not greasy. Drain on paper towels.