In a large bowl, combine the cumin and oregano with half of the garlic and 1 tablespoon of the lemon juice. Stir in the oil and a generous pinch of salt. Add the lamb and onion and let stand for 15 minutes.
Meanwhile, in a blender or mini food processor, combine the tahini with 1/2 cup of water and the remaining garlic and 2 tablespoons of lemon juice and puree until smooth. Add the mayonnaise and puree until creamy. Season the white sauce with salt.
Heat a large griddle or 2 skillets until very hot. Add the lamb and onion and cook over high heat, turning occasionally, until the meat and onion are charred and tender, 6 to 7 minutes. Transfer to plates and serve with the tahini sauce, hot sauce, lettuce, tomatoes, rice and pita.