1. Combine the mushrooms, onion, sherry, and 1 cup broth in a 4- to 5-quart nonstick pan over high heat. Bring to a boil and stir often until liquid is evaporated and mushrooms are very well browned, 20 to 25 minutes; add 1 to 2 tablespoons water, if needed, to prevent scorching.
2. Rinse turkey neck, heart, gizzard, and liver; wrap liver and chill. Add neck, heart, gizzard, and 4 cups broth to pan. Cover, bring to a boil over high heat, and simmer until gizzard is tender when pierced, about 1 1/2 hours. Add liver (or save for other uses) and cook until faintly pink in the center (cut to test), 5 to 8 minutes. Pour mixture through a strainer into a bowl. Discard vegetables. Pull meat from neck; discard bones. Finely chop meat and giblets. If making ahead, cover bowl and meat mixture, and chill up to 1 day.
3. Measure broth mixture and, if needed, add water to make 4 cups. Return broth mixture to the 4- to 5-quart pan and add meat mixture and turkey drippings; bring to a boil over high heat. In a small bowl, blend cornstarch with 1/4 cup water. Stir in enough cornstarch mixture to thicken boiling gravy as desired. Season to taste with salt and pepper.
Nutritional analysis per 1/4 cup.