Season the chicken with the 1 teaspoon of salt and pepper. Heat a large Dutch oven over medium heat and melt 1 tablespoon of the butter. Add the chicken, and saute until well browned, about 6 to 8 minutes on each side. Remove meat from the pan and pour off any fat left in the pan. Lower the heat to medium-low, add the remaining butter. Add the celery, leeks, and carrots, saute for 10 minutes.
Tie the parsley sprigs, thyme and bay leaf with a string and add to the pot along with the chicken and broth. Bring the soup to the boil, adjust the heat to maintain a gentle simmer, cover, and cook for 25 minutes. Remove the chicken and bouquet garni. Skim any excess fat from the surface of the soup with a spoon or a ladle. When the chicken is cool enough to handle remove the meat from the bones and cut into 1-inch chunks; add the meat to the soup along with the rice. Cook until the meat and rice are heated through. Season the soup with salt and pepper and chopped parsley. Divide the soup among warm bowls and serve.
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