First, prepare the giblet broth. Rinse the giblets and neck under cold water. Place them in a medium saucepan along with the quartered onion, chopped celery, parsley sprigs, bay leaf, and 4 cups of chicken or turkey broth. Bring to a boil, then reduce heat, cover, and simmer gently for at least 1 hour, or until the neck meat is tender. Skim off any foam that rises to the surface during simmering.

Once the broth has simmered, carefully strain it through a fine-mesh sieve into a clean bowl or another pot. Discard the solids (giblets, neck, vegetables, bay leaf). You should have about 4-5 cups of rich giblet broth. If you have less, you can top it up with additional broth or water.

While the broth is still warm, you can optionally finely mince the cooked heart and gizzard if you like to include them in your final gravy. The liver can be a bit strong for some, so it's often best to discard it after making the broth, but this is up to personal preference.

In a separate saucepan, melt the butter over medium heat. Once melted, whisk in the flour to create a roux. Cook, whisking constantly, for about 2-3 minutes until the roux is lightly golden and smells nutty. This step is crucial for thickening the gravy and removing the raw flour taste.

Gradually whisk in the strained giblet broth, about a cup at a time, ensuring it's fully incorporated and smooth before adding more. Bring the mixture to a simmer, whisking frequently.

Continue to simmer the gravy, stirring often, for about 10-15 minutes, or until it has thickened to your desired consistency. If it becomes too thick, thin it with a little more broth or warm water. If it's too thin, you can make a slurry with a tablespoon of cornstarch and a tablespoon of cold water, whisk it in, and simmer for another minute until thickened.

Season the gravy generously with salt and freshly ground black pepper to taste. Remember to taste as you go! If you've included the minced giblets, stir them in now.

Serve hot and enjoy the incredible flavor!