1. Rinse turkey neck, heart, gizzard, and liver; wrap liver and chill. Place neck, heart, gizzard, carrots, and onions in a 9- by 13-inch metal pan. Add 1/2 cup broth.
2. Bake in a 475° oven until liquid is evaporated and vegetables, neck, heart, and gizzard are browned, 40 to 45 minutes (30 to 35 minutes in a convection oven); stir several times. Add 1/4 cup broth, vinegar, and sugar to pan; stir to scrape browned bits free. Add liver to pan (or save for other uses). Bake until mixture is richly browned and liver is faintly pink in the center (cut to test), 10 to 15 minutes longer (about 5 minutes in a convection oven); stir several times.
3. Remove from oven; set liver aside. Add 1 more cup broth to pan; let stand until browned bits are soft, about 5 minutes. Scrape mixture into a 4- to 5-quart pan. Add remaining 3 cups broth. Bring to a boil over high heat; cover and simmer until gizzard is tender when pierced, about 1 1/4 hours. Pour mixture through a fine strainer into a large bowl. Discard vegetables. Pull meat from neck and chop; discard bones. Finely chop meat and giblets. If making ahead, cover bowl and meat, and chill up to 1 day.
4. Measure broth mixture and, if needed, add water to make 4 cups. Return broth to the 4- to 5-quart pan. 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 boiling gravy to thicken as desired. Season to taste with salt and pepper.
Nutritional analysis per 1/4 cup.