Heat a large grill pan over medium-high heat until hot, or heat a gas or charcoal grill to medium-high.
Cut the onion half through the root end into 6 wedges, keeping the root end intact.
In a medium bowl, toss the onion wedges with the olive oil, 1/2 tsp. salt, and 1/4 tsp. pepper.
Grill the onion for 5 minutes, then flip over and continue to grill until softened and lightly charred, 4 to 5 minutes.
Season the steak with 1 tsp. salt and 1/2 tsp. pepper and grill, flipping once, until cooked to your liking, about 10 minutes for medium rare (130°F) or 12 minutes for medium (140°F). Transfer the steak and onion wedges to a cutting board, tent the steak loosely with foil, and let rest for 5 minutes.
Toast the rolls cut side down on the hot grill pan until lightly browned.
Combine the cheese, mayonnaise, horseradish, and cayenne in a small bowl and spread it on the bottom halves of the rolls.
Trim the root ends from the onion wedges. Thinly slice the steak across the grain. Divide the onion, steak, and greens among the rolls and serve.