Put the beef in a large mixing bowl. Use a rubber spatula to fold in the horseradish and chives; season with salt and pepper. Hand-form the meat into 8 burgers, cover them, and chill in the fridge while preparing your grill.
Place a large grill pan on 2 burners over medium-high heat or preheat a gas or charcoal grill and get it very hot. Burgers stick to a cold grill, so it's important that you give it plenty of time to heat. Take a few paper towels and fold them several times to make a thick square. Blot a small amount of oil on the paper towel then carefully and quickly wipe the hot grates of the grill; this will create a non-stick grilling surface. Grill the burgers for 8 minutes per side for medium; 7 minutes if you like your meat rare.
When the burgers are just about cooked, put a couple of slices of cheese on top of each and cover the grill for 1 minute to melt the cheese. Remove the burgers to a clean side plate so you have enough room to toast the buns.
Rub the grill rack with the same paper towel as before to clean the small charred pieces, then toast the hamburger buns cut-side down for 1 minute. Serve the burgers with your favorite condiment and any garnish you like, such as bacon, lettuce, sliced tomato, and onion.