Preheat oven to 450 f. Combine squash, 1 tbs olive oil, and 1/4 tsp salt. Toss well to coat. Arrange squash mixture in a single layer on a baking sheet. Bake for 25 mins stirring once.

Heat 1 tbs oil in large Dutch oven over med-high heat. Add mushrooms, sauté 5 mins until until browned stirring occasionally. Transfer mushroom mixture to a bowl and keep warm.

Heat remaining 1 tbs oil in pan. Add onion, sauté 4 mins or until tender, stirring occasionally. Add barley and garlic, cook 1 minute, stirring constantly. Add wine, bring to a boil. Cook 3 mins or until liquid is nearly absorbed. Add broth, remaining 1/2 tsp salt, and pepper, bring to a boil. Cover, reduce heat and simmer for 30 mins or until barley is tender and liquid is absorbed.

Remove from heat, add cheese, stirring until cheese melts. Stir in squash, mushrooms, and thyme. Serve immediately.