Place the rice in a large bowl of cold water. Using your hand, stir to release the starch of the rice. Drain and repeat until the water runs clear, then cover the rice with water again and let it soak for 30 minutes.
Bring a saucepan of water to a boil and add the salt. (This is a lot of salt but it’s the only time the rice is seasoned.) When boiling, add the rice and cook until the rice has doubled in size, 7 to 10 minutes. The rice should be par-cooked: soft around the edges while still firm (not crunchy) in the center. Drain the rice and rinse it to stop the cooking.
In a skillet, melt 1 tablespoon of the butter over medium-high heat. Add the barberries and raisins and sauté for 2 minutes. Set aside.
In a medium bowl, combine 1 cup of the cooked rice and the yogurt and egg yolks.
In a large nonstick saucepan or skillet with a cover, combine the remaining 3 tablespoons butter and 2 tablespoons of the oil over medium-high heat. When the butter has melted, add the rice-and-yolk mixture in a layer across the bottom of the pan. Sprinkle with the barberries and raisins, then top with the remaining cooked rice in a dome shape. With the handle of a wooden spoon, make holes into the middle of the dome and 5 other holes around the center. Cook for 6 minutes, making sure you can see the oil bubbling around the edges of the rice. Drizzle the 1 tablespoon remaining oil over the rice.
Wrap a lid with a kitchen towel and the cover the pan. Turn the heat to low and cook until the rice is done, about 35 minutes.
Remove from the heat and remove the lid from the pan. Carefully place a plate on the rim of the skillet and invert, protecting your hands with oven mitts or pot holders. The rice should slide off the pan.