In a medium nonstick skillet, heat 2 tablespoons EVOO, 2 turns of the pan, over medium-high heat. Add the eggplant and garlic and cook until tender, 12 to 15 minutes. Let cool for 5 minutes. Transfer to a food processor along with the tahini, lemon peel and lemon juice and puree until smooth. Season the baba ghanoush with salt and pepper. Wipe out the skillet.
Meanwhile, in a large bowl, combine the lamb, parsley, coriander, cumin, turmeric, cinnamon, 3/4 teaspoon salt and 1/4 teaspoon pepper. Form into 4 patties.
In the skillet, heat the remaining 1 tablespoon EVOO over medium heat until rippling. Add the burgers and cook, turning once, for 5 minutes each side for medium-rare.
Slather each roll top with the baba ghanoush. Set the burgers and lettuce leaves on each roll bottom and cover with the roll tops.