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preheat a grill to medium high.
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melt the butter in a medium saucepan over medium heat.
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add the flour and mustard powder and cook, stirring, 3 minutes.
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whisk in the beer and bring to a simmer; whisk in the half-and-half until smooth.
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return the sauce to a simmer and cook, whisking occasionally, until thickened, 3 to 4 minutes.
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stir in the cheese and horseradish until smooth; keep warm.
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form the beef into four 6-ounce patties
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season the patties on both sides with salt and pepper.
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lightly brush the grill grates with vegetable oil, then grill the patties until marked on the bottom, about 5 minutes.
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flip and cook until marked and slightly firm, about 3 more minutes for medium doneness.
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remove the patties to a plate and let rest 2 minutes.
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grill the rolls, cut-side down, until lightly toasted, about 1 minute.
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top the bun bottoms with the burger patties, cheese sauce, tomato and chives.