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Melt the butter in a large saucepan over medium heat. Add the flour; cook and stir 30 seconds or until bubbly.
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Add the milk, salt, mustard, and cayenne pepper; bring to a simmer over high heat, stirring frequently. Add the broccoli and red pepper; return to a boil.
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Reduce the heat to low; simmer uncovered 5 minutes, stirring occasionally.
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Add the cheese; stir over low heat just until cheese melts (do not boil).
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Ladle into four soup bowls; top with chives or green onions.