Combine the first 5 ingredients in a large saucepan. Bring to a boil; cover and reduce heat, and simmer 10 to 12 minutes or until the vegetables are tender. Remove from heat and set aside.
Melt butter (1/4 cup plus 2 tbsp) in a heavy saucepan over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually ad 3 1/2 cups milk; cook over medium heat, stirring constantly, until the mixture is thickened and bubbly.
Add the reserved vegetable mixture, remaining 1 1/2 cups milk, shredded cheese, salt, and pepper; cook, stirring constantly, until the cheese melts and the soup is thoroughly heated. Serve immediately.