Step 1Heat the oven to 400 degrees. Line a baking sheet with a silicon sheet or parchment paper. Spoon the grated Parmesan in 1-tablespoon mounds on the sheet and press lightly to flatten. Bake until the cheese is melted and browned, about 7 to 8 minutes. Remove from the heat and cool.
Step 2When the Parmesan crisps are done, place the walnuts on the baking sheet, and bake until toasted and fragrant, about 5 minutes. Remove and cool.
Step 3Shave the asparagus: Lay a stalk flat on the cutting board, holding it at the base. Starting at about where the pale base turns green, use a vegetable peeler to shave the stalk in long, even strips all the way through the tip. When you've gotten about halfway through the stalk, turn it over and start on the other side. When you reach the point that the peeler will no longer shave the spear, rest the spear on top of the flat handle of a wooden spoon to elevate it and take the last 2 or three strips. You can either discard what remains or save it to make an asparagus soup.
Step 4Trim the bottoms of the stems from the mushrooms so they'll sit flat on the work surface. Using a very sharp knife, slice the mushrooms as thin as you can.
Step 5Combine the asparagus shavings, mushrooms and walnuts in a large mixing bowl, and toss gently.
Step 6Place the minced chives in the bottom of a small bowl and cover with the olive oil. Add the lemon juice and one-half teaspoon salt, and whisk until smooth. Pour all but a teaspoon or so of this dressing over the asparagus mixture and toss gently with your hands to lightly coat. Add the rest of the vinaigrette, a little at a time, as necessary. Season with salt to taste.
Step 7Arrange the salad on a platter, and crack the Parmesan crisps over the top in large pieces. Serve immediately.