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Melt the butter in a saucepan over medium heat. Sprinkle the flour over the top, stirring constantly until a thick paste forms, 1 to 2 minutes.
Slowly add the milk to the flour/butter paste in a steady stream, whisking constantly. Continue cooking, whisking constantly until the mixture thickens, about 5 minutes. It should be thick but still pourable. Add the cheese, whisking constantly until it is melted. Add the hot sauce and cayenne. Taste, then add salt and pepper to taste. Add more cayenne and/or hot sauce if preferred.
The sauce will thicken upon cooling. Store leftovers covered tightly in the refrigerator for up to 5 days. Heat over stovetop or in the microwave before serving again.