Parboil the onions: Trim the ends, peel, and cut a vertical slit to the core of each onion. Boil in salted water 10-12 minutes until pliable. Drain and cool.
Separate the layers: Peel the onion layers into shells along the slit. Discard the tough outer skin and tiny cores (chop the cores for the filling). Remove the thin inner membrane from each shell.
Mix the filling: Combine the lamb, rice, tomato paste, pomegranate molasses, parsley, mint, garlic, spices, olive oil, water, chopped onion cores, and salt. Mix until loose and evenly combined.
Stuff: Place about 2 tablespoons of filling on each shell and roll snugly. Arrange seam-side down, packed tightly in a single layer in a deep pot.
Braise: Whisk the butter, tomato paste, pomegranate molasses, and water or broth with salt; pour over the rolls to three-quarters up. Set a heatproof plate on top, cover, and simmer gently 40-45 minutes until the rice is tender.
Rest and serve: Rest covered 10 minutes. Serve warm with garlic yogurt, parsley, and spoonfuls of the reduced sauce.