How to Make It
Trim the stems off the okra; do not cut into the pods. Arrange the okra side by side and thread them in small groups on 2 parallel bamboo skewers. Treat the asparagus the same way. Threat the shiitake caps onto a skewer, 3 or 4 per skewer; the caps should lie flat. Using a sharp knife, make shallow crisscross cuts on both sides of each eggplant half. Transfer all the vegetables to a platter. Brush with the sesame oil and season with salt and pepper.
In a small bowl, whisk the miso with the sake, mirin, sugar and mayonnaise.
Light a grill. When the fire is hot, grill the eggplants, cut side down, for 3 to 4 minutes or until nicely browned. Turn the eggplants and brush with the miso sauce. Grill for 6 to 8 minutes longer, until the bottoms are nicely browned and the flesh is soft. Transfer the eggplants to a platter and sprinkle with 1/2 teaspoon of the sesame seeds.
Grill the okra, asparagus and shiitakes, turning, for about 4 minutes or until nicely browned and cooked through. Transfer the vegetables to the platter, sprinkle with the remaining 1 1/2 teaspoons of sesame seeds and serve.