01 - Pat steak completely dry with paper towels. Season both sides generously with salt and pepper. Heat olive oil in a large skillet over high heat until smoking. Sear steak for 3 to 4 minutes per side for medium-rare doneness, or adjust timing to your preference. Transfer to a plate and rest for 5 minutes before slicing thinly against the grain.
02 - Melt butter with olive oil in a large skillet over medium heat. Add sliced onions and salt, stirring to coat. Cook for 20 to 25 minutes, stirring occasionally, until onions are deeply golden brown and very soft. Sprinkle sugar halfway through cooking to enhance caramelization. Deglaze with balsamic vinegar, stir well, and cook for 1 additional minute.
03 - Preheat broiler or oven to 400°F. Place split rolls cut-side up on a baking sheet. Toast for 2 to 3 minutes until lightly golden and crisp.
04 - Spread mayonnaise on the bottom half of each toasted roll. Layer sliced steak evenly across rolls, followed by caramelized onions and 2 slices of cheese per sandwich. Return to oven for 2 to 3 minutes until cheese is melted and bubbly. Top with arugula if desired, close sandwiches, and serve immediately.