Warm a panini press or heavy skillet over medium-high heat.
Combine the mayonnaise, horseradish, and onion, and spread the mixture over the bread.
Drizzle each slice of beef with a little steak sauce, distributing it evenly. Place two slices of beef onto each of 4 bread slices, and top with evenly divided amounts of arugula and cheese. Add the remaining bread slices, compressing the sandwiches slightly.
Coat the outside of the sandwiches with cooking spray, and heat them in a panini press—or skillet under a heavy weight (such as another skillet)—until they are browned and the cheese melts, 3 to 4 minutes per side.
Tip: To serve these as appetizers, cut each sandwich like a pizza into 6 wedges.