Heat grill to high; lightly oil grates. Season steaks with salt and pepper; grill 4 to 5 minutes per side for medium-rare. Transfer to a plate; let rest 10 minutes.
Meanwhile, grill onion until tender and charred, 4 to 5 minutes per side. Transfer to a plate; separate into rings. (To store, refrigerate steaks and onion, up to 1 day.)
Assemble sandwiches: Thinly slice steaks against the grain. Dividing evenly, spread top halves of bread with cheese, and bottom halves with mustard. Pile steak, onion, and arugula on bottom halves; close sandwiches. Cut each sandwich into quarters, and serve.