Add the oil to a heavy-bottomed Dutch oven over medium heat. When the oil begins to shimmer, add the carrots, onions, and celery. Sauté, stirring occasionally, until the onions are translucent and the vegetables are softened slightly, 5–7 minutes.
Add the garlic, mushrooms, barley (pre-soaked and drained), mixed herbs, and bay leaves; sauté an additional minute.
Pour in the chicken stock and water, scraping up any browned bits on the bottom of the pot. Season with salt and pepper. Bring the soup to a boil, then reduce heat to a simmer. Cover partially with a lid (I crack the lid open about 1/2") and simmer for 20 minutes, stirring occasionally.
Add the cooked chicken. Continue simmering for 25 additional minutes. Check the barley for texture: if you prefer a softer consistency, continue to simmer up to 1 hour. (If the broth gets low, add an additional cup of water.)
Check the seasoning of the soup, adjusting if needed. Serve topped with a sprinkling of Romano cheese, parsley, and a couple grinds of black pepper.