Preheat oven to 350°, or 325° convection. Line a 13" x 18" baking sheet with foil.
Wash the vegetables and trim away any spoiled parts. Cut the vegetables into evenly sized chunks and place them on the foil-lined baking sheet. Sprinkle lightly with salt and a good drizzle of olive oil, up to 3 Tablespoonsful. Mix gently together with your hands. Place the sheet into the oven and roast for about 40 minutes, or until the onions and other vegetables are beginning to take on some roasted color and are quite fragrant, stirring halfway through.
Place the roasted vegetables and any browned stuck-on parts and oil that remain into an 8 quart stockpot. Add the salt, peppercorns, fresh or dried herbs, dried mushrooms, if using, bay leaves, and water. Bring the pot to a rapid boil, and immediately reduce heat to maintain a simmer. Cover, and cooking for 1-2 hours, stirring often. Remove the lid for the last half of cooking.
Taste the stock and adjust seasoning by adding more salt if necessary. Allow the stock to cook slightly until safe to handle.
Strain the stock first through a colander to remove the larger bits, then strain again through a very fine mesh strainer to remove the tiny bits that make it cloudy. Store in the refrigerator for up to 5 days, or freeze for up to three months.