In small saucepan, combine vegetable broth, mirin, sugar and soy sauce, and cook over medium heat, stirring often, just until sugar dissolves. Remove from heat, and whisk in miso until glaze is smoothly blended.
In small bowl, combine vegetable and sesame oils.
Cut eggplants diagonally to make ovals about ¼ inch thick and 3 inches long. Cut squash crosswise into ½-inch-thick slices. Discard mushroom stems and cut caps in half.
Place sesame seeds in small frying pan over medium heat. Cook, shaking pan frequently, until lightly browned, 3 to 4 minutes. Immediately remove from pan to cool.
Preheat electric grill to 350°F, or heat grill pan over medium heat. Brush grill with some oil mixture.
Place vegetables on grill, and brush with oil. Cook, turning once and brushing again with oil, until vegetables are tender, about 2 to 3 minutes on each side for squash, 1 to 2 minutes for mushrooms and eggplant, and 1 minute for baby corn.
Reduce heat to low. Brush half the miso glaze on vegetables and cook for 30 seconds. Turn vegetables, brush with remaining glaze, and cook 30 seconds. Place on serving plate, and sprinkle with sesame seeds.