Remove the skin from the onions. Cut onions and garlic bulbs in half lengthwise. Place ingredients from the clear soup to the onions in a large soup pot and simmer gently for 20 minutes. Add chicken breasts, one chile pepper, the cilantro, and three lemons cut into halves. Simmer until meat is cooked (about 15 min). Remove meat, cool. Skim soup, pour through a fine sieve into another saucepan. Continue to simmer at low temperature. Peel chicken, remove bones, shred into pieces.
Skin the tomatoes, remove the seeds, add to the soup with the chicken and let simmer. Season with salt. Pour the soup into a warmed tureen. Cut the remaining lemon into slices and garnish the soup with them. Chop the second chili pepper. Remove the skin from the avocados, cut into cubes and sprinkle with the juice of one lemon. Offer the chili pepper and avocado separately with the soup.