In a medium saucepan, cook onion in hot oil over medium heat until tender. Stir in rice and vermicelli mix (including seasoning packet); cook and stir for 2 minutes. Stir in water and broth. Bring to boiling; reduce heat. Simmer, covered, for 20 minutes (liquid will not be fully absorbed).
Meanwhile, preheat oven to 425 degrees F. In a large bowl, combine chicken, tomatoes, chile peppers, chili powder, basil, cumin, and black pepper. Stir in cooked rice mixture. Transfer mixture to an ungreased 2-quart casserole.
Bake, covered, about 25 minutes or until heated through. Sprinkle with cheese. Let stand for 5 minutes before serving. Makes 6 servings.