Heat oven to 350° F. Place the prosciutto in a single layer on a baking sheet. Bake until crisped, 8 to 10 minutes. Transfer to a plate and let cool.
Meanwhile, bring a large saucepan of water to a boil. Add the asparagus and cook until tender, 3 to 5 minutes, depending on thickness. Drain and transfer to a bowl of ice water. Drain again and pat dry. Cut the asparagus into 2-inch pieces. Transfer to a bowl. Add the oil, salt, and pepper and toss. Break the prosciutto into pieces and sprinkle over the asparagus.