Cook the spaghetti according to the package directions.
Meanwhile, with a vegetable peeler, shave the asparagus lengthwise into long ribbons. Add the asparagus to the pasta water during the last 2 minutes of cooking. Drain the spaghetti and asparagus and return them to the pot. Add the oil and hazelnuts; toss. Season with the salt and pepper; toss again. Divide the spaghetti among individual plates. With the peeler, shave the Parmesan over the top.
Tip: To toast hazelnuts, place them on a baking sheet in a 350° oven until they're barely browned and fragrant, 7 to 12 minutes. To remove the skins, transfer the hot nuts to a large kitchen towel, fold one end over the nuts, and rub briskly.