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Season ground beef generously with salt and pepper. (There was a whole section in the magazine about making the perfect burger. You'll have to buy the issue if you want to read about it. This is how I've always done it.) Gently (do not over work the meat) form four patties that are 5 inches wide and ¾ inches thick. Make sure to indent the middle with your thumb. Grill burgers to desired doneness.
Meanwhile, melt butter in a saucepan over medium heat. Stir in flour and mustard powder, cooking a minute or so (do not brown). Whisk in beer, bring to a simmer then whisk in half-and-half. Simmer, whisking, until thickened, 4 minutes. Stir in cheese (you can always add more if you think your sauce isn't thick enough) and horseradish.
Toast buns and top burger with cheese sauce, chives and sliced tomatoes.