Snap off and discard tough ends of asparagus. Bring a large saute pan of salted water to a boil. Cook asparagus until just tender but still dark green, 4 to 5 minutes. Transfer to a clean kitchen towel; fold over. Set aside.
Heat half of the butter in a large nonstick skillet over medium-low heat. Break eggs into skillet. Season with salt and pepper, and cook, covered, until set, about 4 minutes.
Place several slices of prosciutto on each of four plates; arrange 4 spears asparagus on each. Brush with the remaining melted butter. Carefully place fried eggs on top, sprinkle with Parmesan and parsley, and serve.