Step 1

Place the contents from 1 giblet package from a turkey or chicken, the neck, 1 peeled and halved medium carrot, 1 halved medium celery stalk, 5 fresh parsley sprigs with the leaves removed, and 2 fresh thyme sprigs in a small saucepan. Add enough cool water to cover by about 1 inch and bring to a boil over medium-high heat.

Step 2

Reduce the heat to maintain a rapid simmer and cook until the giblets are very tender, about 45 minutes.

Step 3

Remove the saucepan from the heat. Fit a strainer over a large liquid measuring cup or heatproof medium bowl. Pour the contents of the saucepan through the strainer. Transfer the giblets and neck to a cutting board; discard the aromatics. When cool enough to handle, pick the meat off the neck and discard the bones. Finely chop the giblets.

Step 4

Add enough low-sodium chicken broth or drippings to the strained giblet liquid until you have 2 1/2 cups total.

Step 5

Melt 6 tablespoons unsalted butter in a large saucepan over medium heat. Add the giblets and neck meat and cook until the butter starts to brown, about 2 minutes.

Step 6

Sprinkle with 1/3 cup all-purpose flour and whisk until the flour is smooth. Continue to cook, whisking, until the roux turns a nutty brown color, 2 to 3 minutes more. While whisking constantly, slowly pour in the 2 1/2 cups liquid. Bring to a simmer. Cook, whisking occasionally, until the gravy starts to thicken and the whisk leaves defined trails, about 3 minutes.

Step 7

Whisk in 1/4 cup whole milk, 1/4 teaspoon kosher salt, and 1/2 teaspoon black pepper. Continue to simmer, whisking occasionally, until the gravy is thick enough to coat the back of a spoon and slowly drip off the end, 2 to 3 minutes. Taste and season with more black pepper and kosher salt as needed.