Discard fat from the top of the can of chicken broth. Heat a very large stockpot. Remove from the heat and spray twice with vegetable oil spray. Toss in the onion and carrot, lower the heat and cover the pot. Cook, stirring frequently, over medium-high, add the chicken and cook until the chicken flesh is browned on both sides, about 5 minutes.
Pour in the chicken broth and water. Add the celery and bay leaves, and bring to a boi. Boil for 5 minutes. As foam accumulates, skim it off and discard. Lower the heat to simmer and add the thyme. Simmer, covered for 2 hours. Add water as necessary to keep the chicken covered with liquid.
Remove the pot from the heat. Remove the chicken and allow to cool, then pick the meat from the bones and reserve for another use. Strain the stock and discard the vegetables and bay leaves. Cool to room temperature. Cover and refrigerate overnight. Lift any congealed fat from the stock and discard. Store for 2-3 days in the refrigerator or freeze for longer storage.
Makes 20-24 cups.