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; saut 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.
Wine note: The 2008 J Lohr Riverstone Chardonnay (California, $1
is a buttery, beautiful partner for this creamy-textured risotto. Tropical fruit notes, ripe pear, melon, and vanilla aromas combine for a wine that will embrace the rich flavors in this dish. --Natalia MacLean