Preheat the oven to 450°F. Use grapeseed oil to grease a baking sheet.
In a large bowl, combine the onion and potatoes. Season with the turmeric, 1 teaspoon salt, and 1 teaspoon pepper, and spread in the center of the baking sheet. Season the chicken generously with salt and pepper, and place it on top of the vegetables. Cover with foil and roast for 20 minutes. Uncover and roast for 10 minutes, or until the vegetables are tender and the chicken is cooked through. Let cool, then bone the chicken and cut the meat into 1-inch pieces.
In a large bowl, whisk together the yogurt, saffron, lemon juice, and zest. Add 1 teaspoon salt and 1 teaspoon pepper. Whisk in the eggs, then transfer 1/2 cup of the marinade to a sealed container and refrigerate. Add the chicken and vegetables to the remainder and stir to coat with the marinade. Cover and refrigerate overnight.
Soak the rice in cold water for 30 minutes, then drain and rinse in cold water until the water runs clear.
In a stockpot, combine the water and 2 heaping tablespoons salt and bring to a boil. Add the rice and return to a boil, uncovered. After 3 minutes, test a grain of rice by breaking it in half. The rice is ready when it’s soft but the center is still opaque and not fully cooked. Drain and rinse the rice under cold water to stop the cooking.
In a large bowl, combine the reserved 1/2 cup marinade with half of the rice. Heat a 10- to 12-inch Dutch oven or cast-iron skillet over medium heat. Add 4 tablespoons of the oil. Spread the rice mixture over the bottom and 1 to 2 inches up the sides. With a slotted spoon, lift the chicken and vegetables from the marinade and spread them evenly over the rice. Spoon 1/4 cup of the marinade over the chicken and vegetables, and sprinkle with the barberries. Season with salt and pepper. Spread the rest of the rice over the chicken and vegetables, and spoon the remaining marinade over the top. Smooth the rice with a spatula, gently working in the marinade. Drizzle with the remaining 2 tablespoons oil.
Cover the pan tightly and cook over medium heat for 8 minutes, then turn down the heat to very low. If you have a flame tamer, put it between the burner and the skillet to disperse the cooking heat. Cook for 11/4 hours. Turn off the heat and let the pie rest, covered, for 15 minutes.
Loosen the sides of the pie with a butter knife and slice into wedges. Or, for a more dramatic presentation, flip the pie onto a baking sheet, then transfer to a platter to serve.