Make the broth: Chop the turkey neck into 3 or 4 pieces with a cleaver. Chop the gizzard in half. Heat the oil in a large saucepan over medium-low heat. Add the neck, gizzard, tail, and heart (do not use the liver) along with the onion. Stir to coat with oil, cover, and cook gently, stirring occasionally, for 20 minutes. The meat will begin releasing lots of juice.
Add 4 cups cold water and the carrot, celery, herbs, and peppercorns. Bring to a boil over medium-high heat, cover, and reduce the heat to maintain a gentle simmer. Simmer until the broth is flavorful, 30 to 40 minutes. Strain the broth and set aside until the fat rises to the top. Skim off and discard the fat. Use the broth immediately or cool and refrigerate for up to 3 days or freeze for up to 3 months.
Make the gravy: Heat the giblet broth until hot. Pour the drippings from the roasting pan into a heatproof measuring cup or fat separator. Allow the fat to rise to the top and then spoon 4 Tbs. back into the roasting pan. Separate and discard the remaining fat from the pan juices. Season the giblet broth with the pan juices, adding only enough to make the broth very flavorful but not too salty. If necessary, add water until you have 4 cups of liquid.
Place the roasting pan over two burners set on medium heat. Sprinkle the flour into the pan and use a flat whisk or wooden spoon to combine it with the fat. Cook for about 2 minutes.
To keep lumps from forming in the gravy, slowly pour about 1/2 cup of the broth mixture into the pan while whisking vigorously to disperse the flour evenly into the liquid. The liquid should thicken quickly and get gluey. As soon as it thickens, add another 1/2 cup or so of broth while whisking. Repeat until the gravy starts looking more like a smooth sauce than glue. At this point, you can whisk in the remaining broth and bring the gravy to a simmer. Add the reserved herb stems and simmer for about 5minutes to develop the flavors. Strain the gravy through a medium sieve, add the sage, thyme, and lemon juice, and season to taste with pepper.