In a bowl, whisk together olive oil, pomegranate molasses, lemon juice, minced garlic, ground cumin, kosher salt, and freshly ground pepper.
Put marinade and lamb* in a resealable plastic bag, seal, and squish to mix. Chill at least 4 and up to 24 hours.
Heat a grill to high (450° to 550°). Grill lamb and onion rings, turning once, until onions are slightly softened and charred and lamb is medium-rare (cut to test), 7 to 13 minutes total. Transfer to a board and let rest 5 minutes. Roughly chop onions. Cut meat from bones and thinly slice.
In a bowl, combine lamb and any juices, onions, 1/2 cup pomegranate seeds, and the mint. Set out a platter with lamb, pita, lettuce, tahini dressing, and the remaining pomegranate seeds for build-your-own sandwiches.
In a bowl, whisk together sesame tahini, lemon juice, extra-virgin olive oil, water, finely chopped flat-leaf parsley, minced garlic, kosher salt, and cayenne.
*Lamb chops vary in size and tenderness, with loin being smaller and more tender but also more expensive.