Heat a large soup pot or Dutch oven over medium-high heat; add 2 tablespoons of the oil. Season the meat generously with salt and pepper. Sear the meat on all sides until well browned; this will take about 15 minutes. Wipe the pan out with a paper towel. Lower the heat to medium, add the oil to the pan. Add the carrot, onion, and celery to the pan and saute until tender, about 10 minutes. Return the meat to the pan with the water. Bring to a boil, adjust the heat to maintain a gentle simmer, cover, and cook for 1 1/2 hours or until the meat is just tender. Add the thyme, barley, and tomato, continue to simmer the soup, covered, for 45 minutes.
Meanwhile, in a medium saute pan, heat the butter over medium-high heat, add the mushrooms and saute until golden, about 10 minutes. Season the mushrooms with salt and pepper and add them to the soup, and simmer for 15 minutes more. Remove the meat from the soup; cut the meat from the bone and dice. Skim any fat from the surface of the soup with a ladle or large spoon. Return the meat to the soup with the parsley. Season the soup with salt and pepper. Serve in warm bowls.
If making the soup in advance, adjust the consistency of the soup when re-heating with water, since the barley has a tendency to absorb the broth.
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