Shape beef into two oval patties; sprinkle with salt and pepper. In the same skillet, cook patties over medium heat for 3-4 minutes on each side or until a thermometer reads 160° and meat juices run clear; drain. Remove and keep warm.
Spread remaining butter over one side of each slice of bread. Place in skillet, buttered side down, and toast until lightly browned.
Combine salad dressing and horseradish. On each of two slices of toast, layer a slice of cheese, a beef patty, and half of the onion and the dressing mixture. Top with remaining toast.