Chop together anchovy fillets and garlic, adding a pinch of salt. Mash into a paste using the side of a knife blade, then transfer to a medium bowl.
Whisk in egg yolks, lemon juice and mustard.
Gradually whisk in olive oil, drop by drop at first, then more steadily. Follow with the vegetable oil until the dressing is thick and glossy.
Whisk in Parmesan. Season with salt, pepper and more lemon juice, if desired. Set aside.
Using a mallet, pound the chicken breasts to about 3 mm thick. Season with salt and pepper.
Marinate the chicken in a bit of prepared Caesar dressing for at least 30 minutes.
Grill the chicken over medium heat for 3-4 minutes on each side until golden and crisp. Set aside.
Brush the bread with olive oil on both sides and grill for about 2 minutes on each side until crispy. Optionally, rub with the remaining garlic clove.
Optionally, lightly brush the cut side of the romaine lettuce hearts with olive oil and grill for 1-2 minutes, until slightly charred but still crisp.
To serve, arrange the grilled lettuce on a plate, top with sliced grilled chicken, add the grilled bread (or cut into croutons), and garnish with shaved Parmesan. Drizzle with the Caesar dressing.