Wash the whole chicken or chicken pieces well. If using a whole chicken, cut off the tail end. Place in a large deep pot (about 7 to 8 quart volume) and fill with enough cold tap water to cover the chicken by 1 or 2 inches. If using 4 pounds of chicken, add about 4 quarts water.
Place over medium-high heat to bring to a boil. Use a sieve to remove the scum that rises to the surface and discard. Add the rest of the broth ingredients. Lower the heat to maintain a simmer, not boil. Partially cover and let cook for about 90 - 120 minutes. If too much liquid is evaporating, add a little more water.
Remove the pot from heat. Using a flat metal strainer, carefully lift the chicken or chicken pieces out of the pot and place in a large bowl to cool slightly. Strain the broth into another pot to catch the vegetables and any debris from the bones. Discard the vegetables. The Parmigiano Reggiano rind can be reserved (either whole or chopped) for the broth and served with the soup; otherwise, discard it. See below how to defat the broth. (At this point, this broth is more accurately labelled a stock. You can let it cool and then freeze to use in other recipes like risotto).
Once the meat is cool enough to handle, use your fingers or two forks to remove the meat from the bones. You can use all the meat to add to the broth or you may decide to reserve some of it to make chicken salad or chicken pot pie. Either dice or shred the chicken and place it in a bowl. Use some of the broth to keep it submerged. Set aside.