Preheat an outdoor grill to medium-high heat (375 degrees F to 400 degrees F/180 degrees C to 200 degrees C). You can also use a stovetop grill pan set over medium-high heat.
For dressing, mince 2 garlic cloves and transfer to a small bowl. Add mayonnaise, 2 tablespoons Parmesan cheese, anchovies, Worcestershire, mustard, and 1/4 teaspoon each salt and pepper; whisk well to combine.
Brush cut sides of romaine, bread slices, and lemons with olive oil.
Lightly oil grill grates (or pan). Place romaine, bread slices, and lemon halves, cut sides down, on grill. Grill lemon halves until lightly charred, 3 to 4 minutes. Grill bread, turning halfway through, until toasted, 2 to 4 minutes. Grill romaine until lightly charred, 1 to 2 minutes. Do not turn romaine halves. (If using a pan, grill in batches.)
For croutons, rub remaining garlic clove over cut sides of toast. Cut toast into bite-size cubes.
Squeeze juice from 1 grilled lemon half. Whisk into dressing.
Arrange romaine on a large serving platter. Sprinkle with remaining 4 tablespoons Parmesan and 1/4 teaspoon each salt and pepper. Top with croutons. Serve with dressing and remaining grilled lemon halves.
