Heat a Dutch oven over medium-high heat. Add 1 tablespoon oil; swirl to coat. Add half of beef; cook 10 minutes, turning to brown well on all sides. Remove beef from pan. Repeat procedure with remaining beef; remove from pan.
Add 1 tablespoon oil to pan; swirl to coat. Add onion, celery, and garlic; sauté 5 minutes. Add tomato paste; cook 1 minute, stirring constantly. Add 1 cup water, scraping pan to loosen browned bits. Stir in beef, stock, barley, thyme, and bay leaves. Bring mixture to a boil; cover, reduce heat, and simmer 1 hour. Add carrot; simmer 45 minutes to 1 hour or until beef is tender. Discard thyme and bay leaves; stir in salt and pepper.
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