Cut 1 lime into 4 to 6 wedges; set aside for serving. Juice as many of the remaining limes as needed to measure 1/4 cup.
Season the chicken breasts with the 1 tsp. salt and 1/2 tsp. pepper. In a large saucepan over medium heat, warm the olive oil. Add the chicken, and cook until browned, about 5 minutes. Transfer the chicken to a plate. Add the garlic and chili and sauté until fragrant, about 1 minute. Add the broth, water, lime juice and oregano, then return the chicken to the pan.
Increase the heat to high and bring the liquid to a boil, skimming off any foam that rises to the surface. Reduce the heat to medium-low, cover partially and simmer until the chicken shows no sign of pink when pierced with the tip of a sharp knife near the bone, about 40 minutes.
Transfer the chicken to a carving board and let cool slightly. Meanwhile, keep the soup at a simmer. Remove and discard the bones, and shred the chicken into bite-size pieces. Stir the chicken into the soup. Adjust the seasonings with salt and pepper.
Put the lime wedges in separate small bowls. Ladle the soup into warmed bowls and serve immediately.
Try adding the following:
1 large white onion, chopped. Add the onion to the pan and sauté until translucent before sauting garlic