Place chicken in a large stock pot and add enough water to cover the chicken. Add the celery, salt, pepper, oregano, and celery salt in a large stock pot, cover, and bring to a boil over medium heat.
Reduce heat and let simmer for 2 to 3 hours.
Remove chicken from the pan and set aside. Use forks (or your hands if you allow the chicken to cool a bit) to pull off small chunks of meat. Remove all of the meat from the chicken and set aside.
Remove and reserve the broth from the pan (or use another large stock pot).
Add 6 cups of broth, the chicken, and the rice back to the pot and simmer, covered, for 15 minutes.
Stir in the carrots and another cup of broth, cover, and simmer for 5 minutes more.
Continue to add broth, cover, and simmer for 5 minutes, until the rice is almost done.
When the rice is getting close and should only need another 5 minutes, add the broccoli and repeat the broth, cover, simmer routine. (I prefer my broccoli still have a bit of crunch to it, but you can add the broccoli sooner if you like it softer.)
Salt and pepper to taste.