Preheat a grill to medium high. Melt the butter in a medium saucepan over medium heat. Add the flour and mustard powder and cook, stirring, 3 minutes. Whisk in the beer and bring to a simmer; whisk in the half-and-half until smooth. Return the sauce to a simmer and cook, whisking occasionally, until thickened, 3 to 4 minutes. Stir in the cheese and horseradish until smooth; keep warm.
Form the beef into four 6-ounce patties, about 3/4 inch thick; press the center of each patty with your thumb to make a 1/2-inch-deep indentation. Season the patties on both sides with salt and pepper. Lightly brush the grill grates with vegetable oil, then grill the patties, indentation-side up, until marked on the bottom, about 5 minutes. Flip and cook until marked and slightly firm, about 3 more minutes for medium doneness. Remove the patties to a plate and let rest 5 minutes. Grill the rolls, cut-side down, until lightly toasted, about 1 minute.
Top the bun bottoms with the burger patties, cheese sauce, tomato and chives. Cover with the bun tops.