If the giblets were included with the chicken, discard the liver and put the rest in a deep, narrow 8-quart stock pot. Remove the breast meat from the chicken and save for another use. Pull off and discard any large pieces of fat from the cavity opening.
Rinse the chicken, especially its cavity, and put it in the pot. Add 3 quarts cold water, plus more if necessary to cover the chicken. Bring to a simmer over medium-high heat and then reduce the heat to maintain a bare simmer. Cook for 30 minutes, skimming off any scum with a slotted spoon or skimmer.
Add the remaining ingredients, cover, and continue to cook at a bare simmer for 2 hours, adjusting the heat and skimming as necessary.
With tongs and a large slotted spoon or skimmer, remove most of the solids, transferring them to a bowl to cool before discarding. Slowly strain the broth through a fine sieve set over another large pot. If there are cloudy dregs as you near the bottom, stop straining and discard them.
Taste the broth; if you’d prefer its flavor to be more concentrated, simmer it until it’s as flavorful as you like. Depending on how you’ll be using the broth, you may want to season it with more salt at this point. If the broth is very fatty, chill it and then remove the solidified fat with a slotted spoon. Refrigerate for up to 5 days, or freeze for longer storage.