Snap the asparagus spears to break off the woody, white ends.
In a large saute pan, heat 1 tablespoon of olive oil over medium-high heat. When the olive oil is hot, gently lay the asparagus in the pan and season with salt and pepper. Cook, rolling the asparagus spears in the pan, for about 4 minutes, or until lightly caramelized on all sides.
Meanwhile, in another large saute pan, melt the butter over medium-high heat. When the butter is melted, crack the eggs into the pan and cook for 2 to 3 minutes, or until cooked sunny side up. Season to taste with salt and pepper.
Lay 5 or 6 asparagus spears on each of 4 serving plates and put 1 fried egg on top of each portion. Garnish with shaved cheese and season to taste with pepper and a drizzle of olive oil.