Place the chicken pieces in a large heavy pot. Add enough water to cover the chicken by about 4 inches (10 cm).
Add one half of the onion, 2 cloves of garlic, the black peppercorns, and the bay leaf to the pot. Bring the water to a boil over medium-high heat.
Reduce the heat to maintain a gentle simmer. Cook for 30 minutes, then use tongs to remove the chicken breasts to a plate. Leave the remaining chicken in the pot and simmer for 1 hour and 30 minutes. Ensure the chicken reaches a safe internal temperature of 165°F (74°C).
While the broth simmers, slice the remaining half of the onion into long strips. Mince the remaining 2 cloves of garlic.
Add the sliced onion, minced garlic, carrots, potatoes, chayote, and tomatoes to the pot. Simmer for 20 minutes.
Stir in the corn, zucchini, cilantro, mint, and salt. Simmer for another 10 minutes until all vegetables are tender.
Shred the meat from the reserved chicken breasts and the other pieces in the pot, discarding the bones and skin. Return the shredded meat to the soup.
Serve the broth in bowls. Plate the vegetables and a scoop of rice on the side, along with fresh lime wedges.