In same skillet, heat remaining 1 teaspoon oil over medium heat and cook carrots, celery and onion 7 to 8 minutes or until tender-crisp. Stir in mushrooms and cook about 10 minutes longer or until most of liquid evaporates and vegetables are tender and lightly browned.
Return chicken and barley to skillet; stir in broth, 2 cups water, thyme, nutmeg, salt and pepper. Heat to boiling over medium-high heat. Reduce heat to low; cover and simmer 30 to 35 minutes or until barley is tender and chicken is no longer pink in center.
Serve pilaf garnished with parsley leaves.