Cook chicken with onion, carrots, celery and salt in boiling water as you would for soup. Set chicken aside. Drain broth into a large stock pot. Discard vegetables.
Add rinsed posole. Cook over medium heat 2 hours.
Remove chicken from the bone and add to posole. Add remaining ingredients.
Simmer slowly, approximately 1 1/2 hours. During cooking, continue to add canned chicken broth and water as necessary.
Stir occasionally to prevent scorching and sticking.
Serve with Red Chile Sauce.
Notes
This is a traditional holiday dish in New Mexico and is usually made with pork. We prefer it with chicken. I buy frozen red chile sauce and prepare it as directed.