Combine the chicken, celery, onion, garlic, parsley, lemon zest, mint sprigs, red pepper flakes and water or broth in a large saucepan. Bring to a boil and skim off any foam. Reduce the heat to medium low and simmer, skimming as needed, 45 minutes, or until the chicken is tender.
Strain the stock through a fine sieve into a large bowl. Transfer the chicken to a plate to cool and discard the vegetable mixture. Return the stock to the saucepan over medium-low heat and simmer until it is slightly reduced and flavorful, about 30 minutes. Add the rice and simmer until tender, about 20 more minutes.
When the chicken is cool enough to handle, remove the skin, then remove the meat from the bones and tear it into bite-size pieces. Add to the stock and cook 3 to 4 minutes. Add the lemon juice, chopped mint, and salt and cayenne pepper to taste; cook 2 to 3 more minutes.