Heat oil in a Dutch oven over medium-high heat. Add carrot and thyme; sauté 1 minute. Add celery, leek, and onion; sauté 1 minute. Add salt, pepper, and chicken; sauté 5 minutes. Add barley; sauté 1 minute.
Add broth and water; bring to a boil. Cover, reduce heat, and simmer 40 minutes. Remove from heat; discard thyme sprig. Stir in parsley and cheese.