Make the white sauce by blending mayo, sour cream/yogurt, garlic, sugar, lemon juice, and black pepper.
Make the spicy rice by toasting rice with butter, olive oil, garlic, tomato paste, and adding chicken broth and various spices. Cook for 20 minutes.
Make the lamb shawarma by marinating lamb with olive oil, allspice, coriander, garlic, paprika, cumin, salt, sumac, black pepper, and turmeric. Cook in a skillet until browned.
Assemble bowls by adding rice as the base, topping with lettuce, tomatoes, cooked lamb, and pita. Drizzle white sauce, hot sauce, and optional barbecue sauce on top.