Grind the saffron threads in the palm of your hand, or in a mortar and pestle (preferred). Mix with 2 tablespoons of warm water in a large bowl. Let sit and allow saffron to bloom and draw out as much color as possible. It should be a deep ruby red.
Zest 1 teaspoon of the orange and set aside.
Bring a large pot of water to a boil and add ¼ cup salt and stir.
Place rice in a strainer or sieve and rinse with lukewarm water, swishing rice around with your fingers to get rid of excess starch. Continue to rinse until water runs clear.
Add rice to boiling water and stir to prevent rice from sticking to the bottom. Cook, stirring occasionally, until rice begins to rise to the top and is tender but still has a slight bite to it, 6 minutes. Drain and rinse with cold water.
Melt butter in a medium skillet over medium-low heat. Cook barberries and golden raisins stirring often, until plumped slightly and warmed through, about 2 minutes. Remove from heat and stir in orange zest and rose water.
Preheat oven to 400°F and place rack in lower third of oven.
Add the egg yolks, yogurt, oil, and salt to the saffron and mix well with spatula. Add rice to yogurt mixture and gently fold in to ensure every grain is coated.
Coat a 10" diameter glass pie dish with oil (glass lets you check on the color from underneath). Add half of rice mixture and use the spatula to gently pack the rice into dish. Scatter half of fruit mixture over and top with remaining rice. Press down again, this time more firmly.
Cover dish tightly with foil and bake until rice on the bottom and around edges is a deep golden brown, 65–75 minutes. Discard foil and let cool 10 minutes.
Loosen the rice around the edges using the point of a knife. Place a large plate on top and invert rice onto plate like a cake. Scatter remaining fruit mixture over top. Slice and serve.