Place the chickens, carrots, onion, and salt in a 12-quart pot. Add enough cold water (about 16 cups) to cover. Bring to a boil. Reduce heat and simmer gently, uncovered, for 1 hour. Skim off any foam that appears.
Transfer the chickens to plates and let cool. Remove the carrots and onion with a slotted spoon and discard. Add the rice to the broth and simmer for 20 minutes.
Meanwhile, shred the chicken meat, discarding the skin and bones. Add the meat and ¼ teaspoon pepper to the broth and heat for 3 minutes.
Scoop the avocado halves from the peels and divide between the bowls (or quarter and divide between the bowls). Ladle the soup over the top. Sprinkle with the cilantro and add a squeeze of lime juice.