Brown the beef: Pat the beef cubes dry with paper towels and season with salt and pepper. Heat the oil in a large pot over medium-high heat. Sear the beef in batches for 3-4 minutes per side until well browned, then remove to a plate. Do not crowd the pot — crowded beef steams instead of browning.
Cook the onion, carrots, and celery: Add the onion, carrots, and celery to the same pot. Cook for 5 minutes, stirring often, until the onion softens, scraping up the browned bits from the bottom — those bits are pure flavor.
Add garlic and tomato paste: Stir in the garlic and tomato paste. Cook for 1 minute until the garlic smells fragrant and the tomato paste darkens slightly.
Pour in the broth: Pour in the beef broth and stir well. Return the browned beef to the pot. Add the bay leaves and thyme, bring to a boil, then reduce the heat to a gentle simmer.
Add the barley and simmer: Stir in the rinsed barley. Cover the pot with the lid slightly open and simmer on low for 45 minutes, until the beef starts to turn tender.
Add the potatoes: Stir in the potato chunks. Cover again and simmer for 20 to 25 more minutes, until the potatoes are tender and the beef is fork-tender.
Season and serve: Remove the bay leaves. Taste the soup and add more salt and pepper as needed. Ladle into bowls and sprinkle with fresh parsley. The soup thickens as it sits — add a splash of broth when reheating if needed.