Remove the giblet bag from the chicken if included and place the chicken in a large stock pot. Add enough water to cover the chicken.
Chop 2 of the carrots and all of the celery into 3 or 4 chunks each and add to the pot, along with onion, salt, pepper, oregano, and celery salt.
Bring the liquid to a boil, reduce the heat and let simmer for 2 to 3 hours. The chicken should be very tender and easily come off the bone when done.
Remove chicken from the pot and set aside to cool a bit. Use forks (or your hands if cool enough) to pull off small chunks of meat. Remove all of the meat from the chicken and set aside. Discard carcass.
Strain the broth from the pot into a large bowl, discarding the vegetables.
Add 6 cups of broth, the shredded chicken, and the rice to the pot and bring to a simmer. Cover and cook for 15 minutes.
Peel and dice the remaining 4 carrots and add them to the pot, along with 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 tender and cooked through. You'll need to taste the rice to check for doneness.
Season with salt and pepper to taste. Serve hot.