Start by making the marinade. In a small bowl, combine the olive oil, red wine vinegar, Worcestershire sauce, lemon juice, garlic and water. Add pepper, to taste.
Put the chicken into a sealable plastic bag and pour in half of the marinade. (Set aside the rest of the marinade.) Refrigerate for 30 minutes, turning chicken over after 15 minutes.
To make the salad dressing, whisk together the mayonnaise and the remaining marinade. Refrigerate until ready to use.
Prepare a hot fire in a charcoal grill or heat a gas grill or broiler. Away from the heat source, lightly coat the grill rack or broiler pan with cooking spray. Position the cooking rack 4 to 6 inches from the heat source.
Remove chicken from marinade (discard marinade). Grill or broil the chicken until browned and cooked through, about 8 minutes a side, or until the internal temperature reaches 165 F. Transfer chicken to a cutting board, cover and let rest 5 minutes before slicing into strips.
In a large bowl, combine the lettuce, croutons and salad dressing. Toss thoroughly. Transfer to plates and top with the grilled chicken and grated cheese. Serve immediately.