Put rice in a large bowl of cold water. Swish with your hands to release starch, then drain. Repeat until water runs clear. In a medium bowl, dissolve 3 tbsp. kosher salt in 2 cups cold water. Add rice and soak at least 1 hour and up to 8 hours.
While rice soaks, microwave 2 tbsp. water until hot (not boiling). Add ground saffron; set aside to steep.
Fill a pot with 3 qts. water and 1/4 cup salt and bring to a boil, covered. Drain rice and add to the pot. Stir once gently. Bring back to a gentle (not rolling) boil, uncovered, and cook rice, skimming off any foam, until softened but still with a bite to it, 4 to 6 minutes. Drain rice in a colander and rinse quickly under lukewarm water. Let sit in colander to drain fully.
Put yogurt in a medium bowl, add 1/4 tsp. saffron water, and stir to mix. Stir in 3 cups drained rice and set aside. Heat another 2 tbsp. water until hot (not boiling). Add 2 tbsp. melted butter; set aside.
Set a 10- to 11-in.-wide dutch oven with a tight-fitting lid over medium heat. Add 3 tbsp. melted butter, swirling it around to cover bottom of pot and up sides by 1 in. With the back of a large spoon, spread rice-yogurt mixture evenly across bottom of pot. This will be the tahdig (crispy rice) layer. Spoon in remaining rice to form a cone, with rice at sides of pot only 1/2 in. deep over tahdig layer. Using handle of a wooden spoon, poke several holes in rice cone (without disturbing tahdig layer) so steam can escape. Drizzle butter-hot water mixture over rice.
Set pot lid on a large clean kitchen towel, fold edges of towel up over lid, and cover pot with lid. Cook over medium heat (to set the tahdig) until steam starts to escape from beneath lid, 10 to 12 minutes.
Turn heat as low as possible and cook rice, rotating pot a few times for even cooking, until rice is tender and tahdig is golden and crisp (check by lifting with a rubber spatula), 50 to 60 minutes.
Meanwhile, soak barberries: Sort through berries, picking out any stems and small stones. Put in a small bowl, add 2 cups cold water, and soak 15 minutes to rehydrate. Drain.
Heat 1 tbsp. melted butter in a small pan over medium-low heat. Add barberries and 1 tsp. reserved saffron water. Cook, stirring, 2 minutes to just warm through. Stir in sugar; remove from heat.
Microwave 1 more tbsp. water until hot but not boiling. Pour into a medium bowl. Add remaining 2 tbsp. melted butter and remaining 1 tbsp. or so saffron water. When rice is ready, add 1 cup rice and 1/3 cup barberries, toss, and set aside.
Gently spoon about 1 cup cooked rice from pot (without disturbing tahdig) and arrange on a platter. Scatter 2 tbsp. berries and a quarter of Saffron Chicken (including onions) on top. Repeat with same amounts of rice, berries, and chicken to form three more layers. Top with saffron rice and barberries. Using a rubber spatula, gently lift out tahdig, break into pieces, and place around edges of rice.
*Find dried barberries (zereshk) at Middle Eastern stores and online. They should be bright red; darker ones are usually old.
Note: Nutritional analysis is per serving of rice.