Light a grill. Using a fork, prick the eggplants in a few places. Grill over high heat, turning occasionally, until the eggplants are very soft and blackened all over, about 35 minutes. Transfer to a baking sheet and let cool slightly. Cut the stems off the eggplants and scrape off the charred skin. Tear the eggplants into long strips and discard the seeds. Transfer to a bowl.
In a small bowl, mix the lime leaves with the lime juice, soy sauce, chile, brown sugar, garlic and lemon zest. Stir 3 tablespoons of the dressing into the eggplant. Arrange the eggplant, avocado, carrot, cucumber, cherry tomatoes and red onion on a platter. Drizzle the remaining dressing over the vegetables. Sprinkle the green onions, mint and cashews over the salad and serve.