On a lightly floured surface, cut dough in half and roll out into two ½-inch-thick rounds. Brush one round generously with clarified butter.
Sprinkle garlic, celery, and onion evenly across dough. Arrange lobster pieces on top. Sprinkle with cheese and parsley. Drizzle half of remaining butter on top.
Repeat with second dough round. Bake 10 to 15 minutes.