Whisk the minced anchovy fillets, garlic, mayonnaise, Worcestershire sauce, Dijon mustard, olive oil, finely grated Parmesan cheese, and lemon juice in a small bowl. Season with salt and pepper to taste. Set aside while you grill the lettuce.
Brush the cut side of the romaine lettuce with olive oil. Sprinkle with salt and pepper. Brush the baguettes with olive oil if using.
Place lettuce on preheated grill (350-400ºF/177-204℃) cut-side down. Gently press down the lettuce to create direct contact with grill grates, ensuring even cooking, more charred and crispy bits. Grill for 1-2 minutes or until slightly charred. Flip the lettuce over and cook for an additional 1-2 minutes. Do the same with sliced bread. (Don't walk away from the grill because it goes fast).
Remove the lettuce and bread from the grill and place them on a cookie tray or large serving plate. Place grilled lettuce on individual plates, drizzle with Caesar dressing, grilled bread or croutons, and extra dressing on the side.
Slice the grilled chicken, and top the salad. Serve immediately for the freshest flavors and textures.