1. Preheat the oven to 425°F. In a large roasting pan, combine the olive oil, shallots, and carrots and toss to coat. Add the beef bones and scraps and roast until browned and the vegetables are lightly caramelized, about 45 minutes, tossing once or twice for even browning.

2. Transfer the beef and vegetables to a large soup pot along with the bouquet garni. Pour off the fat in the roasting pan and set the pan over a burner on medium heat until sizzling. Add the tomato paste and cook until lightly caramelized, about 1 minute. Add 2 cups of the water and cook, scraping up any browned bits stuck to the pan. Scrape the contents of the roasting pan into the soup pot along with the remaining water and mushroom trimmings, if using, and bring to a boil. Skim any scum that rises to the surface. Partially cover and simmer over low heat until the stock is flavorful and reduced slightly, about 3 hours.

3. Strain the stock into a large, heatproof bowl, pressing hard on the solids to extract as much liquid as possible. Let cool, then refrigerate until chilled. Spoon the fat from the surface and, using a paper towel, blot any congealed fat from the surface. Will keep, tightly covered, in the refrigerator for up to 4 days or in the freezer for 3 months. I recommend freezing it in 2-cup containers.