To make the broth, place the chicken, celery, garlic, onion, salt, pepper, bay leaf, and lime in a pot and cover with 4 cups water. Bring to a boil, then reduce the heat to low and simmer uncovered for 30 minutes.
When the broth is done, turn off the heat. With tongs, remove the chicken from the pot and place it in a mixing bowl.
Strain the broth, throw out the vegetables, wipe out the pot, and then pour the broth back into the pot. You should have about 4 cups of broth.
Meanwhile, for the soup, turn on the broiler and place a rack 6 inches from the heating element.
Place the tomatoes seed-side up on a sheet. Place under the broiler and cook for 15 minutes. After this time, add the garlic, onion, and jalapeño to the sheet with the tomatoes, and cook for 10 more minutes. Remove the sheet from the oven. Place the vegetables in a blender.
When the blender contents are no longer steaming, about 10 minutes, puree the vegetables. Scrape out the tomato blend and add to the soup pot with the strained broth. Pour 1 cup water into the blender and swirl around to get any remaining bits and then pour this into the soup pot, too.
Stir into the soup pot the cilantro, cumin, oregano, and rice. Turn the heat up to high, bring to a boil, then turn the heat down to low and simmer uncovered for 30 minutes or until the rice is fluffy and tender.
Meanwhile, when the chicken is cool enough to handle, remove the skin, pick the meat off the bones, and either shred or cut it into bite-sized pieces. Sprinkle the chicken with salt to taste, then add to the soup pot while the rice is cooking. (Since the chicken is cooked, it’s not necessary to add it when you add the rice. You could also make the chicken and broth a day ahead if you prefer.)
Once the rice is cooked, stir in the lime juice. Taste and adjust seasonings, if desired. Serve warm.
Notes
If you want to take a shortcut, you can replace the homemade broth with 4 cups prepared, and stir in 2 cups cooked, shredded chicken.