Chicken Shawarma

Name: Chicken Shawarma
Ingredients: 1 1/2 pounds boneless skinless chicken thighs; 2 tablespoons extra-virgin olive oil, plus additional for oiling the grill grates; Zest of 1 lemon and 1 tablespoon lemon juice; 1 teaspoon ground coriander; 1/2 teaspoon paprika; 1/4 teaspoon ground allspice; 1/4 teaspoon ground cardamom; 1/4 teaspoon ground cinnamon; 1/4 teaspoon ground ginger; 1/4 teaspoon ground nutmeg; 1/4 teaspoon crushed red pepper; 2 cloves finely grated garlic; Kosher salt and freshly ground black pepper; 3/4 pound turnips, peeled and cut into 1/2-inch thick sticks; 1 small red beet, peeled and quartered; 3/4 cup white distilled vinegar; 1 dried bay leaf; 1 clove garlic, sliced, plus 2 finely grated; Kosher salt; 1/3 cup mayonnaise; 1/3 cup yogurt; 3 tablespoons lemon juice (from 2 lemons); 1/3 cup tahini; 8 pita rounds (without pockets); 2 plum tomatoes, diced; 2 cups finely sliced romaine lettuce; 1/2 cup parsley leaves, roughly chopped; Hot sauce (optional)