Scrape the drippings, which will be in gel form on cold chickens or liquidy if you're working with a freshly cooked rotisserie chicken, into a small pot.
Remove the chicken meat from the bones and set bones aside.
Add water to the pot with chicken drippings and place several of the bones (or you can use the whole chicken carcass) in the pot as well. I just use the smaller bones and I don't worry if there is some meat of skin still attached, that just adds to the flavor. Bring to a boil then let simmer for at least 30 minutes but ideally for an hour or more. The longer you can let it simmer, the more flavor you'll get out of the bones but if you're in a hurry, 30 minutes will do the job.
If you want to add additional flavor to your stock, try adding a wedge of onion, a rib of celery and a carrot. You can also spice your broth with herbs that pair nicely with chicken such as rosemary, basil, thyme, parsley, cumin, mustard, and/or cayenne pepper if you like a little heat.
Remove bones from the pot and strain out any extras you added.
In a small bowl, microwave butter and flour for about 30 seconds. Stir and then microwave for an additional 15-30 seconds or until mixture becomes bubbly. Add this to your chicken stock to thick it.
Then stir in chicken bullion. Bring to a boil one more time while stirring, then remove from heat. Shred and add chicken to the gravy.
Serve chicken and gravy over mashed potatoes, rice, biscuits, or egg noodles.