Trim fat and gristle from the chuck roast and cut into ¾-inch cubes. Pat very dry with paper towels and season with salt and pepper; dice the onion, slice the celery, mince the garlic, slice the carrots, rinse the barley, and chop the parsley.
Heat 1 tablespoon olive oil in a 6-quart Dutch oven over medium-high heat. Brown half the beef in a single layer for 3 minutes, turn, and brown 3 to 4 minutes more; move to a bowl and repeat with the remaining oil and beef, keeping the browned bits in the pot.
Lower the heat to medium and cook the onion and celery for 6 to 8 minutes, stirring often and scraping the browned bits off the bottom, until soft and translucent.
Stir in the garlic and tomato paste and cook for 1 to 2 minutes until the paste darkens to a deep brick color, then add the Worcestershire sauce and stir for 30 seconds.
Pour in the broth and water, scraping up any remaining browned bits, and return the beef and its juices to the pot. Add the bay leaves and thyme, bring to a boil, then reduce to a gentle simmer, partially cover, and cook for 1 hour, stirring every 15 minutes and skimming off foam.
Stir in the rinsed barley and sliced carrots and simmer, partially covered, for 45 minutes, stirring every 10 to 15 minutes and scraping the bottom. The soup is ready when the beef breaks apart easily and the barley is tender with a slight chew; thin with up to ½ cup hot water if needed.
Discard the bay leaves, season with salt and pepper to taste, and stir in the parsley. Rest off the heat for 5 minutes, then ladle about 1½ cups into each bowl and serve hot.