Place vinegar in shallow pan and place in the freezer. Allow the vinegar to freeze, approximately 2 hours. Once frozen, scrape with a fork to create a shaved ice texture. Return the vinegar to the freezer until ready to use.
Cut each heart of romaine in half lengthwise so that the root keeps each piece together. Lightly brush the cut side of the romaine pieces with olive oil. Season with pepper. Place the cheese in a shallow pan large enough to lay the romaine in and press the cheese firmly onto the cut side of the romaine so it adheres.
Spray a nonstick griddle or saute pan with vegetable spray and set over medium-high heat. Place the romaine, cheese-side down, in the pan and cook until the cheese turns golden, 1 to 2 minutes. Place the romaine, cheese-side up, onto plates and sprinkle with the vinegar ice. Serve immediately.