Heat grill to medium-high. Place 1 teaspoon oil and all but 2 cloves of the garlic on a piece of heavy-duty foil and close to form a pouch.
Grill the pouch, shaking occasionally, until the garlic is soft, 12 minutes.
Meanwhile, rub the chicken with 1 teaspoon oil and season with ¼ teaspoon each salt and pepper. Grill until cooked through, 5 to 6 minutes per side.
Grill the bread for 1 minute per side.
Peel and finely chop the 2 raw garlic cloves. Add to a large bowl with the anchovies, lemon juice, mustard, Worcestershire sauce, and remaining oil. Toss with the lettuce.
Serve the lettuce with the chicken and Parmesan. Squeeze the grilled garlic cloves from their skins and serve with the toasts.