1. Rinse chayote and cut into 1/2-inch cubes (you can cut through edible seed).
2. Set a 5- to 6-quart pan over high heat. When hot, add oil, onion, garlic, chiles, and cumin; stir until onion begins to brown, 2 to 3 minutes. Add broth, tomatoes (including juice), garbanzos, and the chayote; bring to a boil.
3. Add chicken in a single layer, pushing down to submerge in liquid. Return to a boil, then cover pan tightly and remove from heat. Let stand until chicken is no longer pink in center of thickest part (cut to test), 15 to 17 minutes. With tongs, lift chicken out. Return soup to a boil over high heat; if chayote is not tender, simmer, covered, until tender when pierced. Cut chicken breasts crosswise into thick slices, leaving pieces next to each other; with a wide spatula, transfer each sliced breast half to a wide bowl.
4. Ladle soup into bowls. Distribute avocado and cilantro over chicken and garnish with lime wedges to squeeze over servings. Add salt to taste. Serve with rice to spoon into soup.