Combine squash, 1 tablespoon oil, and 1/4 teaspoon salt; toss well to coat. Arrange squash mixture in a single layer on a baking sheet. Bake at 450° for 25 minutes, stirring once.
Heat 1 tablespoon oil in a large Dutch oven over medium-high heat. Add mushrooms; sauté 5 minutes or until browned, stirring occasionally. Transfer mushroom mixture to a bowl; keep warm.
Heat remaining 1 tablespoon oil in pan. Add onion; saut´´ 4 minutes or until tender, stirring occasionally. Add barley and garlic; cook 1 minute, stirring constantly. Add wine; bring to a boil. Cook 3 minutes or until liquid is nearly absorbed. Add broth, remaining 1/2 teaspoon salt, and pepper; bring to a boil. Cover, reduce heat, and simmer 30 minutes or until barley is tender and liquid is nearly absorbed. Remove from heat; add cheese, stirring until cheese melts. Stir in squash, mushrooms, and thyme. Serve immediately.