In a medium bowl, cover the rice with cold water and stir in 1 tablespoon of the salt. Let soak for 1 hour, then drain.
Meanwhile, in a medium Dutch oven, heat 2 tablespoons of the butter over medium heat. Add the onion and cook, stirring often, until softened, 4 to 5 minutes. Add the chicken and continue to stir until cooked through, 5 to 6 minutes. Transfer to a medium bowl.
To the chicken, add 2 teaspoons of the salt, 1/2 cup of the yogurt, the barberries and turmeric. Stir to combine, then set aside.
In another medium bowl, combine the boiling water and the saffron. Let sit for 5 minutes, then whisk in 1 teaspoon of the salt, 2 tablespoons of the melted butter, the remaining 1/2 cup of the yogurt and the egg.
Preheat the oven to 375° F and grease a non-stick muffin tin with butter. Bring a large pot of water to a boil; season with the remaining 1 tablespoon kosher salt. Add the soaked rice to the pot of boiling water and cook until just tender, 5 minutes, then drain.
To the bowl with the saffron mixture, stir in 3 cups of the cooked rice. You will be making wells for the chicken to go into. Divide the saffron rice between the 12 wells of the muffin tin (2 heaping tablespoons per well) and using a spoon spread up the sides.
Divide the chicken mixture between each well of rice. Top each with the remaining plain rice (you may have a little leftover). Drizzle the remaining melted butter over the rice. Cover the tray with foil and bake, rotating halfway through, until golden, 2 hours.
Let cool slightly, then run a mini offset spatula around the edge of each. Place a tray over the pan, then invert to release the tahchin. Garnish with more barberries and serve.