Tuiles: Place a 12-inch non-stick pan over medium-high heat. Sprinkle 1 tablespoon of the Parmesan on the bottom of the pan. As it starts to melt, use a spatula to form it into three 2 by 5-inch strips. When the cheese starts to bubble and turn golden brown, remove the pan from the heat. Let cool for 4 to 5 minutes; the strips should be pliable. Pick up one end of a strip with a pair of tongs and lay it across the bottom of an overturned bowl or across a wide cylinder so that the tuile hardens into an arc. Set aside and repeat the process with the other strips. Repeat the entire process with the remaining cheese. (Yields 9 Tuiles) White Truffle-Oil Vinaigrette: Blend all the vinaigrette ingredients except their oils in a 2-quart stainless steel bowl. Whisk in the olive and white truffle oils in a slow, steady stream until it is all incorporated and the sauce emulsifies.
Salad: Remove the mushroom caps from their stems. Clean the tops with a damp cloth or towel. Peel the stem using a vegetable peeler; discard the scraps. Slice the mushrooms as thinly as possible, and place in a 2-quart stainless steel bowl. Add the remaining vegetables. Add the vinaigrette and toss well.
To serve: Divide the salad among 6 chilled plates. Stand a Parmesan tuile over each salad. Handle gently and serve immediately.