Bring a wide shallow pan of salted water to a boil—the water should be salty enough that you can taste it.

Blanch the asparagus for about 1 minute until al dente. Immediately transfer to a bowl of iced water to stop the cooking. Drain and dry thoroughly on paper towel.

Wipe out the pan and return it to medium-high heat. Melt half the butter, then add the asparagus and sauté gently for about 1 minute. Season with sea salt and pepper and grate over about 1 tablespoon of Parmigiano-Reggiano. Swirl or toss to coat. Divide the asparagus evenly between 4 serving plates.

Wipe out the pan again and melt the remaining butter over medium heat. Once sizzling, reduce the heat to low and gently crack in the eggs, taking care not to break the yolks. Cook until the whites are just set and the edges are lightly golden.

Carefully lift each egg onto a bed of asparagus using a spatula. Drizzle any remaining butter from the pan over the top. Finish with more grated Parmigiano-Reggiano and a crack of black pepper. Serve immediately.