01 - Pat steak dry and season both sides with salt and pepper. Heat olive oil in a large skillet over high heat. Sear steak for 3-4 minutes per side for medium-rare, or until desired doneness. Transfer steak to a plate and let rest for 5 minutes. Slice thinly against the grain.
02 - In a large skillet, melt butter with olive oil over medium heat. Add sliced onions and salt. Cook, stirring occasionally, until onions are very soft and golden brown, about 20-25 minutes. Sprinkle with sugar halfway through for deeper caramelization. Deglaze with balsamic vinegar, stir, and cook 1 more minute. Remove from heat.
03 - Preheat broiler or oven to 400°F. Place open rolls on a baking sheet and toast lightly for 2-3 minutes. Spread mayonnaise on the bottom half of each roll. Layer sliced steak, caramelized onions, and 2 slices of cheese on each roll. Return to oven to melt cheese, about 2-3 minutes. Add arugula or spinach if desired, close sandwiches, and serve hot.