For the rice: Melt the butter over medium heat in a large Dutch oven. Add the turmeric and cumin and cook until fragrant but not browned, about 1 minute. Add the rice and stir to coat. Cook, stirring frequently, until the rice is lightly toasted, about 4 minutes. Add the chicken broth. Season to taste with salt and pepper. Raise the heat to high and bring to a boil. Cover, reduce to a simmer, and cook for 15 minutes without disturbing. Remove from the heat and allow to rest until the water is completely absorbed and the rice is tender, about 15 minutes.
For the sauce: In a small bowl, combine the mayonnaise, yogurt, sugar, vinegar, lemon juice, parsley, and 2 teaspoons black pepper. Whisk to combine. Season to taste with salt.
To serve: Return the entire contents of the chicken bowl (chicken, marinade, and all juices) to the skillet. Cook over medium-high heat, stirring occasionally, until heated through. To serve, divide the rice, lettuce, tomato, and toasted pita bread evenly among four to six plates. Pile the chicken on top of the rice. Top with the white sauce and hot sauce. Serve immediately, passing extra sauce at the table.
