Make Caesar dressing by combining the first 9 ingredients in a small bowl. Refrigerate until needed.
Preheat an indoor or outdoor grill to medium-high. Remove lose outer leaves from each romaine heart. Cut each romaine heart, lengthwise, in half. Brush the cut side of each romaine heart with olive oil; season with salt and pepper. Set the romaine, cut-side down, on the grill and cook 1 to 2 minutes, or until lightly charred. Set a romaine heart, cut-side up, on each of 6 salad plates. Drizzle each romaine heart with Caesar dressing. Garnish lettuce with lemon slices, shaved or grated Parmesan, and croutons, if using.
Romaine hearts are the centre portion of the head. They are sold in that form in bags in our produce department.