Preheat the oven to 400°F.
Chop the carrots, celery, and onions into large chunks. Arrange chicken and vegetables in a single layer on a baking sheet. Roast for about 1 hour, turning once until the meat and bones are evenly browned.
Combine the bones and vegetables in a large stockpot, along with the pan drippings. Add enough water to cover the ingredients by a few inches. Bring the water to a rapid simmer over high heat on the stovetop, then turn the heat down to the lowest setting possible.
Keep the broth at a low simmer for at least 4 hours. Check the pot occasionally, skimming off any foam that collects on the surface and adding additional water as needed to keep the ingredients covered. Once cooked remove bones, celery, onion and carrots by straining with a sieve. Store the broth in glass containers. The broth will keep refrigerated for up to 7 days, or frozen for up to 3 months.
Reheating: Pour out as much broth as you'd like and reheat it gently on the stovetop. Season with salt.
If you're using a carcass that has already been cooked you can skip roasting step and move right into adding the carcass and vegetables to the water.