Bring stock to a boil in a large pot over high heat. Add chicken legs; return to a boil. Reduce heat, cover, and simmer for 10 minutes, skimming away foam that forms on the surface. Add carrots, onion, celery, cilantro, garlic, and mint.
Cover pot and cook until chicken is no longer pink at the bone and the juices run clear, about 30 minutes more. Remove chicken from the pot; discard skin and bones. Shred meat using 2 forks.
Return chicken meat to the pot and season with salt and pepper. Add rice. Let chicken and rice warm up, about 5 minutes. Remove from heat and pour soup into serving bowls. Top with avocado and serrano peppers and squeeze in lime juice.