How to Make It
Preheat the oven to 350°. In a small saucepan, combine the sugar with 1 teaspoon of water. Cook over low heat, swirling the pan, until the sugar melts and an amber caramel forms, 4 minutes. Whisk in the black vinegar. Remove from the heat and let cool to room temperature. Whisk in the oil and soy sauce.
In a bowl, mix the miso with the mirin. In a nonstick skillet, heat 3 tablespoons of the oil. Add 1 eggplant half, cut side down, and cook over moderate heat until deep-golden, 3 minutes. Transfer the eggplant cut side up to a baking sheet. Repeat with the remaining 3 tablespoons of oil and eggplants.
Spread the miso mixture on the cut sides of the eggplants. Roast for 15 minutes, until the eggplants are very tender. Transfer to a platter.
In a large bowl, toss the tomatoes with the 1/3 cup of dill, 1/4 cup of shiso, the scallions and two-thirds of the dressing; season with salt.
Spoon the salad over the warm eggplants and drizzle with the remaining dressing. Garnish with dill and shiso and serve.
Black vinegar is slightly sweet with a rich, malty flavor. It's typically made from fermented rice, wheat, barley and sorghum. It's available at Asian markets and from amazon.com.