Prepare the chicken: cut into four portions (two breasts and two leg portions), removing any visible fat and cutting off the skin from the neck and the tail openings. Put all the pieces into a very large pan, along with the carcass (this will add extra flavour), then pour in the water and add a teaspoon of salt.
Bring to the boil, then immediately turn down the heat to simmer. Find a large spoon and skim off the scum and fat that rises to the surface. Make sure the water is just letting up small bubbles and keep removing the scum until the soup is clear. This will take about 5-10 minutes and if you do not look after it at this stage, the soup could taste bitter and look cloudy.
Now add all the other ingredients, except the green part of the leek. Turn up the heat slightly to bring it back to simmer and carry on removing any froth or scum.
When the soup is clear again, turn the heat down as low as possible, place the green part of the leek on top, and put the lid on the pot. Simmer for 2½ hours, but remove the chicken breast portions after 1 hour (leave in the legs).
Lift out all the large solids from the pan, then carefully strain the broth using a large sieve. Press down all the vegetables to extract as much of their juices as you can, then discard them. There should be approximately two litres of broth.
If you want to remove the fat floating on the top, take three sheets of kitchen paper and lay them on the surface then remove and throw away. Taste the broth and add more salt and pepper if needed.
Shred the breast meat into small strips, add to the broth and re-heat. Add chopped parsley and serve the soup very hot.