Brush steaks with balsamic vinegar. Sprinkle liberally with kosher salt and black pepper.
Grill, covered, turning often, until it is cooked to your preference (140 for medium rare, 155 for medium well, and 165 for toast).
Remove and let rest for 10 minutes. Slice thinly across the grain on a bias. Drizzle slices with balsamic vinegar.
Place one slice of cheese on the bottom of half of the bread slices. Place some steak strips, enough to cover the cheese, on the slice, then top with another slices of bread. Repeat with remaining bread slices. Eat any remaining steak.
Brush sides of bread facing out with melted butter. Set aside.
Heat a grill pan or griddle over medium high heat for 4-5 minutes. Add sandwiches, working in batches if needed, until browned on both sides and cheese is melted. Serve warm.