For the dressing: combine the lemon juice, Dijon, and garlic. Add the olive oil in a steady stream while whisking in constantly. Stir in the cheese. The dressing may be made in advance, covered, and refrigerated until serving. Stir or shake well before using.
Preheat the grill to high heat. Lightly brush both sides of the romaine halves with olive oil or spray with an olive oil mister (I’ve also brushed with the dressing). Place the romaine, face down, on the hot (uncovered) grill for 1-2 minutes or until light golden grill marks appear. Flip, and grill for another minute or two. Watch the romaine closely. The outside of the lettuce should be slightly caramelized while the insides are still cool and crisp.
Remove from the grill, and brush the cut sides of each romaine half with the dressing. I like to use a pastry brush to work some of the dressing into the crevices. Sprinkle with a little extra Parmesan, croutons, or your favorite Caesar salad toppings. Serve immediately with extra dressing, to taste.