Slice lemons into 1/4-inch-thick rounds. Place in a bowl and drizzle with 1 tablespoon olive oil. Heat your grill to medium-high. Place lemon rounds on grill and cook until well-browned. Set aside.
Slice Romaine heads in half lengthwise, keeping leaves intact. Brush flat (cut) side with olive oil. Place on grill (cut side down) and cook 1 to 2 minutes (without moving). Place on a serving plate and drizzle with Caesar Dressing (recipe below).
With a peeler, shave shards of the cheese over the top. Sprinkle with Black Olive Croutons (recipe below) and arrange lemon rounds on top.
3 anchovy filets, rinsed, roughly chopped
3 tablespoons grated Parmigiano-Reggiano cheese
2 tablespoons lemon juice
2 large egg yolks
1 garlic clove
1/4 to 1/3 cup extra-virgin olive oil
In the bowl of a food processor, pulse together all ingredients except olive oil. Then with the motor running, add 1/4 cup olive oil. Taste and add more oil if necessary. If too thick, add water to loosen.
Heat oven to 350°. Spread olive paste on each bread slice. Cut bread into 1/2-inch cubes. Place on a rimmed baking sheet and bake until toasted, about 10 minutes. Stir croutons and bake another 5 to 6 minutes. Remove and let cool to room temperature.