Shawarma-style grilled chicken wraps

Ottawa has a large Lebanese population, and shawarma is arguably THE post-bar snack, rivalled perhaps only by poutine and pizza. Last week I got a craving for one and decided to try a homemade version. I consulted my Lebanese friends and quickly established that zataar spice was a must. I picked up a 1 lb bag at a specialty deli - it's the most spice I've ever received for $4. I wasn't able to pry a garlic sauce recipe out of anyone so took to the Internet. It seems like garlic, lemon, olive oil and salt are the main ingredients, but the stuff in takeout stands seems much more white in colour than what I got - maybe they're using lard? Anyways, I used extra-virgin olive oil in my recipe but found the taste off-putting so I think regular olive oil is the way to go. If anyone has spent time in a Lebanese kitchen and has suggestions for me, I'm all ears...(Update: It was suggested to me that canola oil or another vegetable oil is best for garlic sauce. I haven't tried this yet thought I would share the feedback.)

Serves four.

To make chicken: 

Combine 1 cup greek yogurt, the juice of 1/2 lemon, 1/4 cup zataar spice, 2T dried oregano, 1T dried parsley, 1 tsp paprika, 1/2 tsp each of cumin, garlic salt and pepper. Pour over 4 chicken breasts and marinate overnight. Grill or bake, then slice lengthwise into thin strips. 

To make garlic sauce:

Put the bulbs from one head of roasted garlic and one head of raw garlic in a blender. Add juice of 1/2 lemon and a generous amount of salt. Turn on blender and while mixing, slowly add 3/4 cup olive oil (not extra-virgin). I ended up adding 2T mayo to try to cover the taste of extra-virgin.

To assemble wraps:

Layer slices of chicken with pickled turnips (also from specialty deli), slices of pickle, tomatoes, cucumber and lettuce on a whole-wheat pita. Top with tahini and garlic sauce, then roll and secure with aluminum foil. 

Serve with a lettuce, tomato and roasted chickpea salad tossed with a simple oil, white vinegar and lemon juice vinaigrette.