In a 4-quart saucepan, bring chicken broth to a boil. Reduce heat; add the potatoes, carrots, celery, onion, salt, pepper and thyme. Cover and simmer for 15 minutes, or until vegetables are tender.
Meanwhile, melt butter in a medium saucepan over medium-high heat. Whisk in flour until smooth. Gradually whisk in milk. Bring to a boil and stir for 2 minutes, or until thickened. Reduce heat and add cheese, stirring until melted. Stir cheese mixture and shredded chicken into broth. Cook and stir until heated through.
Ladle into serving bowls. Sprinkle with chopped parsley.