For the dressing, add all ingredients except for the oil and cheese to a mixing bowl. While whisking, slowly drizzle in the oil. Mix in cheese and reserve.
Rub steak with cut side of garlic clove and dress with olive oil, rosemary, salt and pepper.
Preheat grill, griddle or cast-iron skillet to medium-high heat. Add meat and cook 10-12 minutes turning occasionally. Let meat rest then thinly slice against the grain.
Brush about 1/3 of the dressing on the bottom halves of the toasted bread. Top with sliced steak. Toss lettuce with remaining dressing then top steak with greens and a little shaved cheese. Set bun tops in place and serve.