In large bowl, stir together barley and broth. Cover with vented plastic wrap, microwave on high 20-25 minutes or until most of the liquid is absorbed, stirring once.
Meanwhile, in large skillet, heat 1 Tbsp olive oil on med-high. Add chicken; sprinkle with 1/4 tsp each salt and pepper. Cook 4 minutes or until golden, stirring once. With slotted spoon, transfer chicken to bowl.
To skillet, add carrot, onion, and thyme. Cook 3 minutes, stirring.
Add garlic to skillet, cook 30 seconds, stirring, then add mushrooms and water, scraping up browned bits. Cook 2 minutes; remove from heat.
Stir in chicken and barley; transfer to shallow 3 qt baking dish.
Toss broccoli with 1/8 tsp salt and remaining oil; arrange on top of barley mixture.