Step 1Combine the water, chicken, onion and garlic in a stockpot. Bring to a boil over medium-high heat. Reduce the heat to low and simmer for 1 1/2 hours, skimming the top occasionally.
Step 2Bring a large saucepan of water to a boil over medium-high heat. Add the carrots, potato, chayote, green beans and corn. When the water returns to a boil, reduce the heat to medium-low and simmer until the vegetables are almost done, about 15 minutes. Add the zucchini and continue to cook until the vegetables are very tender, about 10 minutes. Drain the vegetables and set aside.
Step 3Remove the chicken pieces from the broth and shred into bite-sized pieces, discarding the skin and bones.
Step 4Strain the broth into a clean saucepan. Boil over high heat until reduced to 10 cups, about 20 minutes. Add the chicken, vegetables, cilantro, rice and salt.
Step 5Cook over medium heat just until heated through, about 6 to 8 minutes. Add the avocado.
Step 6Serve in bowls with the lime wedges and chopped onions alongside.