1. Heat charcoal or gas grill to 400°F. Sauté onions in 1 tablespoon olive oil over medium-high heat until soft and caramelized.
2. Grill steak for 4 to 6 minutes per side until internal temperature reaches 145°F. Let stand 5 minutes. Slice on the bias into very thin strips.
3. Combine remaining 3 tablespoons olive oil with pinch of salt and pepper. Use pastry brush to coat 1 side of each bread slice. Top olive oil side of 4 slices bread with 1/4 cup steak sauce, 2 slices cheese, 1/4 onions, steak and 2 more slices cheese. Close sandwiches with olive oil side of bread inside.
4. Heat large skillet over medium-low. Melt butter in skillet and place sandwiches in pan. Cook until bread is golden brown and toasted; gently flip sandwiches over and toast other side until cheese is melted.