Line a small rimmed baking sheet with parchment paper. Lay the prosciutto slices in a single layer on the parchment, without touching. Bake until crispy and fragrant, flipping halfway through, 18 minutes. Cool completely on the tray, then remove to a plate.
Preheat a grill pan to medium heat. Remove the parchment from the prosciutto tray and place the asparagus on the tray. Drizzle with the pesto, season with the salt, and toss well to coat. Place the asparagus on the hot grill pan and cook, turning once, until the asparagus has nice golden grill marks and is tender when pierced with the tip of a knife, about 6 minutes. Transfer the asparagus to a platter, crumble the prosciutto over the top, and break the Parmesan shavings over the prosciutto. Serve warm.