In a large saucepan or Dutch oven, place broth, onion, tomato, garlic, and green chiles. Bring to a boil over high heat. Reduce heat to medium-low; cover and simmer for 10 minutes or until vegetables are tender.

Meanwhile, melt butter in a medium saucepan over moderate heat. Remove from heat and stir in flour. Return to heat and cook for 3 minutes, stirring constantly. Stir in 3-1/2 cups of the hot milk, adding 1/2 cup at a time. Cook and stir about 6-7 minutes until smooth and thickened. Remove broth mixture from heat and stir in the milk mixture, about one fourth at a time. Return to heat and add remaining 1-1/2 cups milk, salt, pepper, and cheese, stirring constantly until cheese is melted and soup is well heated. Do not boil.

Serve immediately, or let cool and refrigerate with plastic wrap directly on surface of soup. Cover and refrigerate for up to 2 days. Reheat soup over low heat just until hot. Garnish as desired. Serves 8.