Tip:Removing the chicken’s skin gives you a less fatty broth that won’t need as much skimming.
Using a small sharp knife and your fingers, remove the skin from the chicken and discard it.
Rinse the chicken well and put it in a large (at least 8-quart), heavy-duty pot or Dutch oven. Add enough cold water to submerge the chicken (about 5 quarts). Cover the pot, with the lid slightly ajar. Bring to a boil over high heat and then reduce the heat to maintain a very gentle simmer. Cook, occasionally skimming off any foam that accumulates on the surface, until foam no longer rises, about 30 minutes.
Add the carrots, celery, onion, 1-1/2 Tbs. salt, and 2 tsp. pepper and simmer until the vegetables start to soften and the chicken is completely cooked through, about 20 minutes.
Using tongs and a slotted spoon, transfer the chicken to a large rimmed
baking sheet. Let cool for 10 minutes; meanwhile, continue simmering the
broth, partially covered. Using your fingers, pull the meat from the
bones and shred it into bite-size pieces; discard any gristle or fat.
Set aside the shredded chicken.
Return the carcass to the broth and simmer, partially covered, until the vegetables are completely soft and the flavor has intensified, about 30 minutes more. If at any time the water level drops below the solids, add water to cover and return to a simmer.
Remove the carcass from the broth and discard. Strain the broth through a fine sieve set over another pot or a bowl large enough to hold the broth. Gently press on the solids with a large spoon to squeeze out any remaining broth.
The broth can be refrigerated for up to 3 days or frozen for up to 3 months.