Heat the oven to 425°F. Heat the oil in a roasting pan in the oven until hot enough to shimmer, but not smoke. Add the bones to the pan and roast, turning occasionally, until they turn a deep brown color. The roasting time varies with the type of bones. Beef bones may take 45 minutes, while chicken bones might take only 25. Transfer the bones to a tall stockpot (set the roasting pan aside for the vegetables), add 3 quarts water (it should cover the bones by 1 inch), and bring to a simmer over low heat. Simmer 4 hours, skimming fat and impurities that come to the surface occasionally.

Add the onion, celery, and carrots to the residual oil in the roasting pan; return to the oven and cook until they are golden brown, about 20 minutes. Stir in the tomato paste, and cook 2 minutes more. Remove the vegetables with a slotted spoon and set them aside. Pour off any fat that remains in the roasting pan (this is called degreasing the pan), then deglaze the pan by pouring in some cool water, and scraping up any browned bits with a wooden spoon.

After the stock has simmered 4 hours, transfer the roasted vegetables (called mirepoix) and the deglazed juices to the pot with the bones, along with the sachet. Simmer the stock 2 more hours. Strain the stock, and season with a tiny bit of salt, if desired.