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