For the stock:
(Yields 2½ quarts)
4 Tbs. olive oil, divided
3 lbs. turkey necks, drumsticks, or wings
2 cups peeled and chopped carrots
4 cups chopped yellow onions
2 cups chopped celery
1 cup chopped shallots
1 cup chopped cremini mushrooms
1 cup chopped maitake mushrooms
1⅓ cups good-quality Madeira
4 sprigs thyme
2 sprigs sage
1 sprig rosemary
5 qts. plus 2 cups water, divided
For the gravy:
1 qt. stock
10 Tbs. unsalted butter, diced and kept cold, divided
3/4 cup all-purpose flour
1 cup good-quality Madeira
Fine sea salt and freshly ground white pepper, to taste
For the stock:
Place an oven rack in middle of the oven. Preheat the oven to 425°F.
Rub the turkey parts in two tablespoons of the olive oil. Arrange them in a single layer in a roasting pan and roast until brown and crisp, about three hours.
Heat the remaining two tablespoons of olive oil in a large heavy-bottomed stockpot over medium heat. Add the carrots and cook, stirring occasionally, for 10 minutes, until golden brown. Add the onions and cook, stirring occasionally, for an additional 10 minutes, until golden brown. Add the celery and cook, stirring occasionally, about eight minutes, until golden brown. Add the shallots and mushrooms and cook, stirring occasionally, for 10 minutes, until golden brown.
Add the Madeira, scraping the bottom of the stockpot to release any browned bits, and simmer for five minutes to burn off the alcohol.
Add the roasted turkey bones to the stockpot. Add two cups of water to the roasting pan, scrape the bottom to release the browned bits, and add them and the water to the stockpot. Add the five quarts of water and bring to a simmer. Reduce the heat to low and simmer the stock uncovered for about three hours, until rich in flavor.
Strain the stock through a fine mesh sieve. Discard the bones and vegetables. You should have 2½ quarts of stock. Cool to room temperature. Cover and refrigerate until ready to use. The fat will rise to the top as it chills.
For the gravy:
Skim the fat off the top of the stock. Transfer one quart of stock to a heavy-bottomed medium saucepan and bring to a simmer over low heat. Turn the heat off, cover the saucepan, and leave it on the stove.
In a separate saucepan, heat six tablespoons of the butter over medium heat. Add the flour and stir with a whisk to form a thick, smooth paste. Cook, stirring constantly, for about eight minutes, until golden brown. Stir in the Madeira and cook, stirring constantly, for five minutes to burn off the alcohol.
Pour in the warm stock and vigorously whisk until it is incorporated. Cook over medium heat, stirring frequently, until a sauce consistency is achieved, 20 to 25 minutes. Season to taste with salt and pepper. Stir in the remaining four tablespoons of cold butter one piece at a time. Strain through a fine mesh sieve. Add more stock if too thick. Serve.
Note: The remaining stock can be refrigerated for three days or frozen in a tightly covered container for up to three months.