Combine the meat, squeezed onion, parsley, garlic, allspice, cinnamon, cumin, salt, and pepper. Mix just until combined, because overmixing makes the kebabs dense. Chill for 15 minutes.
Divide the mixture into 8 portions. With wet hands, press each portion around a skewer into a flat 5-inch log.
Grill over medium-high heat for 4 to 5 minutes per side, until charred and cooked to 160°F.
For the sauce, whisk the tahini, lemon juice, garlic, and salt. It will thicken and look stiff at first. Whisk in the cold water one tablespoon at a time until smooth and pourable.
Serve the kebabs drizzled with tahini sauce, with pita, tomato, cucumber, and red onion.