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Add the butter to a small pot over low heat. When melted, quickly whisk in the flour to make a smooth mixture. (It will resemble a paste.)
Slowly pour in the milk, whisking continually, to make a smooth consistency. Whisk while the milk mixture heats up until you see bubbles around the edges. Simmer for about 2 minutes, or until you can see that the milk has thickened.
Remove the pot from the heat.
Stir in the cheese and salt and whisk until the cheese melts into a thick sauce.
Serve warm with prepared broccoli.