The day before serving, combine water, giblets, carrot, celery, onion, and thyme in a 3-quart saucepan. Bring to a simmer and cook for one hour. Remove the giblets and let cool slightly. Strain the broth and reserve in the refrigerator. Finely chop the giblets and neck meat (I find that a fork works well for this). Refrigerate.

To make the gravy, melt butter in a 3-quart saucepan over medium heat. Whisk in flour and cook the roux for a few minutes to eliminate the starchy taste. You want this roux to be blond in color. Add the reserved broth, whisking. Increase the heat to medium-high, and stir in giblets and chopped egg. Add the cream and taste for seasoning. Add salt and pepper to taste gradually–now is not the time to get the gravy too salty. Cook longer until thickened, if needed. You can always thin the gravy with chicken broth or more cream if it becomes too thick.