Cut chicken in serving-size pieces; place in kettle. Add hot water to cover. Add salt, celery and onion. Simmer, covered, until tender, about 3 hours. Add carrots the last 30 minutes of cooking.
Drain broth from chicken. Skim fat from broth, saving ¼ c. fat for gravy. Strain broth; save carrots. Measure 2 c. broth for gravy. Pour remaining broth back into kettle