Season the beef. Pat the cubed beef dry on all sides with a paper towel, then season with 2 tsp Kosher salt and ½ tsp black pepper.
Heat the pan. Heat a Dutch oven or other large pan (with tight fitting lid) over medium-high heat until very hot, about 3-5 minutes. Add 2 Tbsp neutral oil, then add the beef and cook, stirring occasionally, until browned on all sides, about 6-8 minutes. Work in batches, if needed, so as not to overcrowd the pot. Use a slotted spoon to transfer the beef to a large plate.
Sauté the vegetables. To the same pot, add diced carrots, onion, and celery, then cook, stirring occasionally, about 5-7 minutes, until tender. Add 2 cloves of chopped garlic, stir, then cook about 1 minute more. Transfer the sautéed vegetables to a bowl, then set aside.
Add liquid. Add 2 Tbsp tomato paste, then use a spatula to stir. Once cooked down, add 10 cups of chicken stock (or a combination of chicken stock and water) and 1 Tbsp balsamic vinegar. Return the beef (and any juices) to the pot. Bring to a boil, then reduce the heat to a simmer. Add 4 sprigs of thyme and 1 bay leaf.
Simmer. Cover the pot with a tight fitting lid, then cook for 1½ - 2 hours, or until the beef is fork tender.
Add the vegetables and barley. Use a spoon to skim any fat off the top of the soup, then return the sautéed vegetables to the pot. Add ¾ cup pearled barley, cover, then cook for 25-30 minutes, or until the vegetables are tender and the barley is fully cooked.
Finish, then serve. Remove the fresh thyme and bay leaves, then add the juice of 1 lemon and stir. Taste and adjust seasoning as needed, adding more salt or pepper to taste. Serve immediately with chopped parsley or dill for garnish.