Use this stock in Martha's Stuffed Pumpkin recipe.
Preheat oven to 450 degrees. In a large roasting pan, combine turkey gizzards and neck with white onions and garlic. Drizzle with olive oil and toss to coat. Spread in an even layer and roast until meat and vegetables are very brown, tossing occasionally, about 1 hour.
Transfer contents of roasting pan to a large pot. Add carrots, parsley, mushrooms, sage, and enough water to just cover (2 to 3 quarts). Bring to a boil, then reduce heat and simmer 1 hour. Strain stock through a fine sieve lined with four layers of cheesecloth; discard solids. (At this point, stock can be cooled completely and refrigerated up to 1 week or frozen up to 3 months in an airtight container; thaw in the refrigerator before using.)
Heat olive oil in a saucepan over medium. Saute red onion until completely soft and caramelized, stirring occasionally, about 12 minutes. Add strained stock and the thyme and cook at a bare simmer for 20 minutes. Strain stock again through a fine sieve and discard solids. Season with salt and pepper. (Stock can be allowed to cool and then refrigerated in an airtight container up to 1 day; reheat over medium before serving.)