Remove eggs from the refrigerator at least 1 hour before you intend to use them; they must be at room temperature for this dish.

Rinse the asparagus well, then clean the white part of each one by scraping it with a knife. Arrange all asparagus in a bunch, with tips even. Tie the bunch together, toward the bottom, with thread. Then, with a knife, even off the ends.

Place the bunch of asparagus, standing with tips up, in a tall stockpot or asparagus cooker. Add enough cold water to cover the white parts, at the lower end of the asparagus, then cover with pot and place it on the heat. Bring the water to a boil, add coarse salt, reduce the heat, and simmer for 20 minutes or less, according to the size of the asparagus, without removing the lid.

Remove the pot from the heat and transfer the asparagus to a chopping board. Untie the bunch and cut off the hard white ends.

Heat the butter in a large frying pan. When it is melted, add the asparagus and sauté for 6 or 7 minutes, turning gently with a wooden spoon several times. Sprinkle the grated Parmigiano, salt, and freshly ground black pepper over the asparagus, then cook, still turning gently, for 1 minute more.

Leaving the sautéing butter in the pan, transfer the asparagus from the frying pan to a large serving dish and arrange in a ring, with all the tips facing the center of the dish. Keep the serving dish in a warm place, near the stove if possible.

Heat the butter left over from cooking the asparagus over low heat. Break one of the eggs, and carefully pour the white into the pan with the asparagus butter, saving the yolk in its half-shell. Repeat with the remaining eggs, keeping each white separate from the others in the pan.

When all the egg whites are arranged in the frying pan, sprinkle with salt, then place a yolk on top of each white, taking care not to break the yolk. Sprinkle each yolk with pepper, then cover the frying pan and let simmer very slowly for about 4 minutes.

Remove the pan from the heat, arrange the eggs over the asparagus on the serving dish, and serve immediately, sprinkled with a little freshly ground black pepper.