Set oven to 425°F / 220°C (or 400°F if using convection or your pans run hot).
Arrange carrots, celery, onion, and garlic on a large rimmed sheet pan. Nestle in chicken leg quarters, drizzle with olive oil, and season lightly with salt. Toss vegetables to coat and spread in a single layer.
Roast 35–45 minutes, flipping halfway, until chicken skin is golden and vegetables are browned.
Tip: If things darken too quickly, reduce heat to 400°F or tent with foil.
Move roasted ingredients to a large 8-quart stockpot. Deglaze the roasting pan with 1–2 cups hot water (or wine + water), scraping up brown bits, and add this liquid to the pot.
Add bay leaves, peppercorns, and herbs. Pour in enough cold water to cover by 1–2 inches.
Bring to a gentle simmer over medium heat — avoid boiling. Skim off foam and fat as needed. Simmer 2–3 hours, uncovered.
Optional: After 45–60 minutes, remove chicken, pick off the meat to save for soup, and return bones and skin to continue simmering.
Remove large solids with tongs, then pour stock through a fine-mesh strainer (lined with cheesecloth for clarity) into a clean pot or bowl.
Cool quickly in an ice bath, then refrigerate. Once chilled, remove the solidified fat layer. Season with salt when using.