Melt butter in a large stock pot over medium-low heat, and cook the mushrooms until they are browned and the butter is clear, 1 to 1 1/2 hours.
Remove the mushrooms, coarsely chop them, and set aside. There should be about 1 cup of butter left in the stock pot; whisk the flour into the butter, and gently cook over low heat until the flour mixture turns mahogany brown in color, about 20 minutes.
Whisk in the chicken broth; bring the mixture to a simmer to thicken the stock.
Pour the turkey drippings into a saucepan, then cook and stir the onions and celery in the drippings over medium-low heat until the onions begin to turn brown, about 20 minutes. Stir the drippings and vegetables into the thickened stock. Bring the gravy to a gentle boil, reduce heat, and simmer for about 20 minutes to blend the flavors. Stir in the chopped mushrooms, then season to taste with salt, black pepper, and cayenne pepper.