Prepare the Rice: Parboil the rice in salted water until it is almost fully cooked but still slightly firm. Remove the rice from the strainer and run cold water over it to cool it down and stop further cooking.
Cook the Chicken: In a large pot, heat water until it starts to simmer. Add the chicken breasts along with 1 teaspoon of salt and a dash of pepper. Let the chicken cook until it's thoroughly done, which should take around 15 to 20 minutes. After it cools, shred the chicken into bite-sized pieces and set it aside.
Make the Saffron Yogurt Mixture: In a large mixing bowl, whisk together the egg yolks, yogurt, bloomed saffron (with its liquid), minced garlic, turmeric, ½ teaspoon of salt, and a pinch of black pepper. Add the parboiled rice to this mixture and gently mix until the rice is fully coated with the yogurt and saffron.
Assemble the Tahchin Cups: Preheat the oven to 350°F (175°C) and generously spray a muffin tin with cooking oil. Spoon a thin layer of the saffron rice mixture into each muffin cup and press it down firmly. Add a layer of shredded chicken over the rice, then top with more rice to cover the chicken, pressing down each layer to form a compact cup.
Bake the Tahchin Cups: Place the muffin tin in the oven and bake for about 45-50 minutes until the rice is golden and crispy on the edges.
Prepare the Zereshk (Barberries): While the rice cups are baking, rehydrate the barberries by soaking them in 2 tablespoons of boiling water for about 10 minutes. In a small pan, melt the butter, then add the barberries, sugar, and a pinch of salt. Sauté the mixture for 2-3 minutes until the barberries are plump and sweetened.
Serve: Once the Tahchin cups are crispy and golden, remove them from the oven and let them cool slightly. Garnish each cup with a spoonful of sautéed barberries and serve hot.