Preheat the oven to 550 degrees fahrenheit (or the highest setting).
Break the tough ends off the asparagus or trim the Belgian endive and slice it lengthwise into 8 wedges.
Place the mushrooms in a gratin or other baking dish with the asparagus or endive, the onion, garlic, and thyme. Toss with the olive oil and sprinkle with salt and pepper. Roast the vegetables, turning a few times to ensure even browning, for about 20 minutes, or until tender when pierced with fork.
Transfer the vegetables and their cooking juices to a salad bowl and add the vinegar and a little more salt and pepper if needed. Toss well, taste for seasoning, and add the greens. Toss well. Scatter the Parmigiano on top and serve at once.
Recipe courtesy of The Rose Pistola Cookbook, Reed Hearon and Peggy Knickerbocker, Broadway Books/Random House 1999