Season ground beef generously with salt and pepper. Gently (do not over mix the meat) form four patties that are five inches wide and 3/4 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, for approximately four 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.