In large bowl, combine minced garlic with all spices (cumin, coriander, paprika, turmeric, cinnamon, chili powder, salt, pepper). Add yogurt, olive oil, and lemon juice. Whisk into thick paste.
Add thinly sliced chicken thighs and toss to coat completely. Cover and refrigerate at least 1 hour, preferably overnight.
For garlic sauce: Add minced garlic and salt to food processor. Pulse until very finely chopped, almost paste-like.
With processor running on low, drizzle in vegetable oil in thinnest, steadiest stream possible over 2-3 minutes to create stable emulsion.
Once oil is incorporated and mixture is thick and creamy, add lemon juice and cold water gradually while processing to adjust texture. Sauce should be smooth, white, and fluffy.
Transfer garlic sauce to container and refrigerate at least 1 hour before serving.
Heat large skillet or grill pan over medium-high heat. Add marinated chicken in single layer and cook 6-8 minutes, turning occasionally, until fully cooked (165°F) with browned, slightly charred edges.
Warm pita bread in dry pan for 30 seconds per side or wrap in foil and heat in 350°F oven for 5-10 minutes.
Assemble wraps: lay warm flatbread, add lettuce and tomatoes, spoon hot chicken on top, drizzle generously with garlic sauce, sprinkle with fresh parsley.
Fold tightly or serve open-faced with pickles or fries on the side.
Notes
Marinate chicken at least 1 hour, overnight for best flavor penetration.
For garlic sauce: slowly drizzle oil to create stable emulsion. If sauce breaks, add lemon juice and water, blend smooth, then slowly resume adding oil.
Chicken thighs stay more moist than breasts. If using breasts, reduce cooking time by 1-2 minutes.
Let garlic sauce rest in fridge for 1 hour before serving for best flavor and texture.
For dairy-free, use coconut yogurt or skip yogurt and add extra lemon juice and olive oil.