Make marinade by whisking together oil, lemon juice, soy sauce, Worcestershire sauce, mustard, brown sugar, pepper, salt, and garlic in a small bowl.

Place steak in a large resealable bag and pour in marinade. Seal bag, removing as much air as possible. Place in refrigerator and let steak marinate for at least 4 and up to 8 hours.

In a small bowl, whisk together butter, garlic, chives, thyme, parsley, and salt to make the garlic-herb butter. Set aside.

Ignite all burners on grill, close lid, and preheat for 5 minutes. Clean and oil grilling grate.

Brush onions with oil and place on grill along with steak. Cook onions until softened and well charred on each side, about 5 minutes per side.

Cook steak until it is deeply browned on both sides and an instant-read thermometer registers 125°F when inserted into thickest part of meat (about 3 to 5 minutes per side).

Transfer steak and onions to a cutting board as each is done. Let steak rest for 5 minutes, then slice against the grain on a bias. Remove skewer from onions and separate into individual rings.

Liberally butter the cut side of each piece of bread with garlic-herb butter. Place bread on grill, cut side down, and toast until lightly browned, 1 to 2 minutes.

Transfer bread to a platter and top half of slices with steak, onions, and cheese.

Turn off one burner on grill and place topped bread slices on cool side of grill. Cover and cook until cheese has melted, about 1 minute.

Transfer back to platter and top each sandwich with arugula and the top slice of bread. Serve immediately.