Heat grill pan over high heat.
For the salad, in a small bowl combine olive oil, garlic, a few pinches of salt and about 15 grinds pepper. Cut lettuce heads in half lengthwise and brush with olive oil mixture.
Grill lettuce until nicely blackened and wilted. Grill in stages if your grill pan isn't large enough to grill in one stage. Transfer lettuce to a serving platter.
Squeeze lemon over the lettuce. Sprinkle shaved Parmesan over the top, drizzle with the anchovy oil and finish with parsley.