Preheat the oven to 425°. Trim and peel the broccoli stems. Cut the broccoli lengthwise into long stalks, with 1 or 2 florets each. Cut any remaining florets into 2-inch pieces. Spread the broccoli on a large, rimmed baking sheet. On another large, rimmed baking sheet, put the shiitake on one side and the onion slices on the other. Generously brush all the vegetables with olive oil, season with salt and pepper and roast for about 20 minutes, until just tender and lightly browned. Transfer the broccoli to a platter. Drizzle the balsamic vinegar over the mushrooms and onion rings and stir to coat the vegetables. Top the broccoli with the shiitake and onion and serve.