Eggplant and Mushroom Tahcheen

Name: Eggplant and Mushroom Tahcheen
Ingredients: 2 1/4 cups white basmati rice; 1 1/4 tsp saffron strands; A pinch of sugar; 3 tbsp freshly boiled water; 3 medium eggplants (around 1 1/4 lb), cut lengthways into 1-inch slices; Sea salt and black pepper; Sunflower oil; 3 tbsp butter; 1 medium onion, finely chopped; 3 garlic cloves, crushed; 4 Portobello mushrooms, sliced into large chunks; 1/2 tsp turmeric; 1/2 tsp allspice; 1/4 tsp cinnamon; 1/4 tsp cayenne pepper; 1/2 tsp cumin seeds; 2 egg yolks; Scant cup Greek yogurt; 1 tbsp butter; 2 tbsp barberries (see Cooks' Note); 1/4 cup + 1 tbsp pistachios, roughly chopped; 1 tbsp sugar