Roast the bones: Heat oven to 500°. Put bones into a roasting pan large enough to hold them in a single layer and roast until lightly browned, about 1–1 1⁄2 hours. Add carrots, onions, and leeks to the pan and spread them evenly around the bones. Roast the bones and vegetables until they are deeply browned, about 45 minutes more.
Deglaze the pan: Transfer bones and vegetables to a 15–20-qt. stockpot. Pour off and discard any fat in the roasting pan and place pan over 2 burners on the stove over medium heat. Add 3 cups water to pan; begin scraping up any browned bits from bottom of pan with a wooden spoon. These caramelized morsels of concentrated juice, called the fond—literally, the foundation—will enrich the stock. Simmer for 3 minutes; transfer liquid to pot of bones. Add bouquet garni and tomato paste. The paste will give the stock a deeper flavor and color. Cover bones with 6–8 qts. cold water; set pot over medium-high heat. Starting with cold water encourages the proteins and fats contained in bones to rise to the surface in large pieces, where they can be skimmed and discarded.
Simmer the stock: When the first bubbles begin to appear on the surface of the liquid, reduce heat to medium-low and maintain a very gentle simmer; a bubble should rise to the surface about once per second. Simmering slowly prevents the fat and impurities from being churned back into the stock and clouding it. The strength and concentration of your demi-glace will be determined by the length of time the stock simmers. For the minimum amount of extraction, it should simmer for at least 6–8 hours, but we recommend 12–24 hours for a richer, more gelatinous sauce. Check every few hours and add more cold water if necessary so that bones are always covered.
Skim the fat: Skim fatty froth from surface of stock with a ladle every 5–10 minutes during first hour of cooking to prevent it from clouding stock. After first hour, skim the stock every 30 minutes or so.
Strain the stock: When the stock is ready, set a chinois (a fine-mesh conical sieve) or a fine metal sieve over a clean 8-qt. pot. Strain stock through sieve into the pot. Tap edge of sieve with a wooden spoon to loosen any solids that impede the straining of the stock, but do not force liquid through. Discard bones, vegetables, and bouquet garni. The bones may be reused to make a lighter, secondary stock with fresh vegetables and aromatics, called remouillage and used for sauces and soups. The stock should yield 4–5 qts. If storing stock for another use, you can cool it quickly by placing the pot in a sink half filled with ice water. Once it's cooled, skim the surface again to remove any fat. Transfer the stock you don't plan to use right away to storage containers and refrigerate. Stock will keep refrigerated for up to 1 week or frozen for up to 6 months. To transform the stock into demi-glace, proceed to next step.
Reduce the sauce: Traditionally, the stock for demi-glace was thickened with a roux, but modern chefs have shunned thickeners in favor of reducing stock to a pure, more syrupy consistency. Simmer stock over medium-high heat, skimming occasionally, for 4–5 hours until reduced to 2 cups. Refrigerate for up to 2 weeks or freeze for up to 6 months.