Place vegetables and garlic on foil-lined 10x15-inch pan. Drizzle with 1/4 cup oil; sprinkle with thyme, salt and pepper. Bake for 30 minutes or until tender, turning occasionally. Cut into bite-sized pieces.
Cook Angel hair according to package directions. Drain, reserving 1 cup of pasta water. Return pasta to pot.
Add vegetables, cheese and remaining 1/4 cup oil to pasta. Toss, adding reserved pasta water to desired consistency. Transfer to serving platter.