Martha prefers cooking with smaller eggplants -- they're sweeter, have fewer seeds, and don't have to be salted before cooking. Here, they're topped with yogurt, a spicy herb sauce, and a colorful garnish.
Heat broiler with rack 6 inches from heating element. In a large heavy-bottomed saucepan, heat 2 inches safflower oil to 360 degrees. Working in batches, add eggplant and cook, turning once, until golden brown and very tender, about 4 minutes per side. Transfer eggplant to a paper towel-lined baking sheet and let drain.
In the bowl of a food processor, combine parsley, cilantro, turmeric, garlic, chile, and olive oil. Season with salt, then blend until finely chopped. Place eggplants cut-side up on a rimmed baking sheet; season with salt. Brush with herb mixture and broil until deep golden brown, about 2 minutes.
In a small bowl, combine yogurt and lime zest and season with salt.
Serve eggplants garnished with yogurt mixture, sliced chile, lime wedges, sesame seeds, and pomegranate seeds.