Yogurt Marinated Grilled Chicken Shawarma

Name: Yogurt-Marinated Grilled Chicken Shawarma
Ingredients: 1/2 cup whole-milk yogurt; 1/4 cup olive oil; 4 cloves garlic, minced; 1 lemon, juiced; 1 teaspoon ground cumin; 1 teaspoon ground coriander; 1 teaspoon ground paprika; 1 teaspoon onion powder; 1/2 teaspoon ground allspice; 1/2 teaspoon ground cardamom; 1/2 teaspoon ground cinnamon; 1/2 teaspoon ground turmeric; Kosher salt; 2 pounds boneless skinless chicken thighs, pieces halved if large; 1 large onion; 1/2 cup garlic cloves, peeled (about 2 heads); 1/2 teaspoon table salt; 1 3/4 to 2 cups vegetable oil; 2 tablespoons lemon juice, from about 1 lemon; 2 tablespoons ice water; 4 to 6 lavash or pita; 2 cups shredded lettuce; 1 onion, halved, thinly sliced and seasoned with sumac; 2 Roma tomatoes, halved and sliced; 3 dill pickles, cut lengthwise into 6 spears; 18 ounces frozen French fries, cooked