Cut romaine hearts in half, lengthwise. Rinse and pat dry with paper towels.
Preheat grill to high.
Prepare Caesar Salad Dressing.
Add to a blender: lemon juice, garlic, white wine vinegar, Dijon mustard, coconut aminos, olive oil, anchovy filets, salt, and egg yolks. Blend and scrape down sides as necessary. Transfer to a covered container and refrigerate.
Shave about ¼ cup of parmesan with a vegetable peeler, and reserve.
Lightly brush fronts and backs of romaine halves with olive oil. You could also spray them with a Misto sprayer.
Lay romaine on the preheated grill and cook until grill marks form and you can begin to smell the romaine, but not so long that they wilt. It took mine less than a minute. Flip romaine and grill the other side, for an equivalent time. Every grill is different, so you really need to watch these.
Remove to a serving platter, cut side up. Garnish with shaved parmesan, drizzle with Caesar dressing, and season to taste with salt and pepper.
You can serve the heads whole or chopped and tossed in a salad bowl. If serving whole, remove the base/root end of the lettuce heads.