Heat grill to medium low. Cut each head of romaine in half lengthwise, keeping the bottom on so the leaves remain attached. Dry well with a paper towel and then use a basting brush to baste each of the 6 halves with avocado oil lightly.
Place lettuce heads on the grill, turning often with a tong, cooking for a total of approximately 4 minutes. Do not over cook letting the leaves burn or become too wilted. Immediately remove from the grill and allow to cool for 5 minutes.
Combine the olive oil, garlic, lemon juice, vinegar, Worcestershire, mustard, anchovy paste, and salt in a blender, and mix into well blended.
Place the grilled romaine halves on a plate, and pour dressing on top. Add the cheese and pepper, and serve immediately.