In a 4-quart stock pot, melt butter and add oil. Brown the giblets on all sides.
Add onion, celery, carrots, bay leaves, salt, and pepper. Sauté a few minutes then add enough water to cover and fill the pan. Bring to a boil, then reduce to simmer.
Cover and simmer 1 ½ to 2 hours. The vegetables will just about fall apart after this much time.
Remove from heat and pour through a fine-mesh strainer, squeezing all the juices out of the vegetables and giblets.
Set aside and reserve stock for gravy.
For the gravy
Pour drippings from the pan the turkey was roasting in into a fat separating measuring cup. Allow to cool 5-10 minutes. Then pour drippings into saucepan and place over medium-high heat. Bring it to a simmer.
Whisk together 1 tablespoon corn starch and ¼ cup cooled stock until there are no clumps.
Slowly pour cornstarch mixture into simmering drippings, whisking to incorporate. The gravy will immediately begin to thicken.
While whisking, add stock in a slow stream until the desired consistency is achieved. As the gravy sits it will continue to thicken so additional stock may be needed before serving. Reserve any extra stock for soup - it's too good to throw out!
Season gravy with salt and pepper to taste and serve warm.