Light a grill. Brush the shiitakes, eggplant, zucchini, bread and cheese with olive oil. Grill the shiitakes, eggplant and zucchini over a moderately high fire, turning occasionally, until tender and charred in spots, 8 to 10 minutes; transfer to a platter. Grill the bread until toasted, 3 to 4 minutes; transfer to the platter. Grill the cheese, turning once or twice, until softened and charred in spots, about 2 minutes.