Rinse the chickens and pat dry with paper towels.
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; let cool. Remove and discard the carrots and onion. 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 pepper to the broth and heat for 3 minutes. Scoop the avocados into individual bowls and ladle the soup over the top. Sprinkle with the cilantro and squeeze on the limes.
Recommended: 12-quart pot
To Freeze: Omit the avocados, cilantro, and limes. Let the soup cool, then ladle into large resealable bags, filling each one halfway. Store for up to 3 months.
To Reheat: Thaw overnight in the refrigerator or thaw partially in the microwave. Warm in a covered saucepan over medium-low heat for 20 minutes. Scoop the avocado into bowls, ladle in the soup, and garnish with the cilantro. Serve with the lime wedges on the side.