Slice the eggplants into rounds, salt generously, and let sit for 30 minutes. Rinse and pat completely dry.

Heat 3 tablespoons of the oil in a large pan over medium high heat. Fry the eggplant slices in olive oil until golden on both sides. Transfer to a paper towel lined plate.

Heat the remaining olive oil in a pan over medium heat. Saute the onion until softened, then add the ground beef and cook until browned.

Stir in the tomato paste, turmeric, salt, pepper, and cinnamon. Cook for 3-5 minutes so the tomato paste releases its color and the spices release their flavor. Add the water and simmer until most of the liquid has cooked off. Stir in the barberries and walnuts and remove from heat.

Rinse the basmati rice several times until the water runs clear. Bring a large pot of water to a boil and season generously with coarse salt.

Add the rice and cook for 8 minutes, until soft on the outside and firm in the center. Drain in a colander and rinse with cool water to stop the cooking.

Preheat the oven to 350F. Brush a 9x9 baking pan with oil.

Whisk the yogurt, oil, eggs, and salt in a large bowl until combined. Stir in the bloomed saffron, then gently fold in the parboiled rice.

Spread half the rice mixture evenly in the pan. Top with the ground beef filling, then layer the fried eggplant over the beef. Cover with the remaining rice and pat the top smooth.

Cover tightly with aluminum foil and poke holes in the foil with a toothpick. Bake for 1 hour and 30 minutes, until the bottom is golden brown and a toothpick inserted into the rice comes out clean.

For the topping, melt the butter with the oil in a pan over medium heat. Add the barberries and stir to coat. Stir in the sugar and rose water, then add the almonds and pistachios. Cook for 1 to 2 minutes until the barberries are glossy.