Melt one tablespoon of butter in a large skillet over medium heat. Add the giblets and brown on all sides.
Add the celery, garlic, onion, thyme, and bay leaf to the saucepan and saute for 3-5 minutes.
Once the ingredients are simmering, add water and cover for 1-2 hours. If using turkey giblets, adjust the amount of water appropriately since they are larger. I used smaller chicken giblets in this recipe.
Strain the stock with a fine-mesh strainer and scoop the giblets out using a slotted spoon. Set the stock aside and finely chop the giblets.
Using a different skillet (preferably cast iron), melt the remaining 3 tablespoons of butter. Once the butter melts, add in the flour and whisk until fully incorporated.
Gradually add the strained broth into the butter-flour mixture. Once that is fully mixed, add the pan drippings. Start with 2 cups and add more to achieve desired consistency. Add chicken bouillon and salt and pepper to taste.
Add the chopped boiled egg and giblets and simmer while stirring for 2-3 minutes. Add half and half or milk. Adjust seasonings to taste.
Serve warm over chicken or turkey and mashed potatoes!