Melt the butter in a large saucepan over medium heat, then add the neck, giblets, onion, celery, salt, pepper, and optional dried thyme or sage; cook for about 5 minutes. Stir until the giblets and vegetables are well browned and the pan smells deeply savory.

Pour in the chicken broth, bring it to a boil, then lower the heat and simmer for 30 to 40 minutes. The broth should smell aromatic and look slightly fuller in color before you lift out the neck and giblets with a slotted spoon.

After the turkey or chicken finishes cooking, transfer it to a platter and tent it with foil, then add its pan juices to the broth. Scrape the browned fond from the roasting pan into the liquid, then spoon off excess fat floating on top.

Bring the gravy to a boil and reduce it by about one quarter, then strain it through a fine-mesh sieve and return the liquid to the saucepan. The strained gravy should look clear, deeply colored, and free of celery or onion pieces.

Whisk in 1/2 tsp of glucomannan, remove the pan from active heat, and let the gravy stand for 10 minutes; whisk in the remaining glucomannan if a thicker gravy is desired. The gravy will continue to thicken as it cools and should coat a spoon smoothly without becoming stringy.