1. Quarter the chicken, removing the wings and the back (you can reserve these for making stock) and keeping the bones intact. Cut the thighs from the legs and cut the breasts in half through the ribs; you will have eight pieces. Pat the chicken dry, then season with salt and pepper.
2. Heat the oil in a 6-quart heavy-bottom lidded casserole and brown the chicken pieces over medium-high heat, turning them over so that they brown evenly, about 15 minutes.
3. Remove and reserve the chicken. Pour off all but about 1 tablespoon of excess oil from the pan. Add the leeks, fennel, garlic, rosemary and spring onions and saute over medium heat for about 2 minutes.
4. Return the chicken to the pan, along with any accumulated juices, and add 2 1/2 cups of water. Bring to a boil, then reduce to a simmer and cover and cook until the chicken is done, about 20 minutes. Stir in the carrots during the last 15 minutes of cooking.