Snap off the tough ends of the asparagus; cook in boiling salted water until just crisp-tender, approximately 4 to 5 minutes for thick stalks. Remove from heat and remove asparagus with a slotted spoon; drain on several thicknesses of paper toweling until ready to serve. Remember that the asparagus will continue to cook from their own heat as they stand and cool, so if you like very firm asparagus, time them accordingly. Divide cooked asparagus among 4 individual serving plates. Sprinkle with grated Parmigiano-Reggiano cheese.
In a medium-size frying pan over medium heat, heat the butter until just hot enough to sizzle a drop of water. Break and slip eggs into the pan; reduce heat to low and cook approximately 2 to 3 minutes or until whites are set but yolks are still runny. With a spatula, remove the eggs to the plates you will serve them on and place on top of the asparagus (be careful not to break the yolks). The eggs should be hot enough to slightly melt the grated parmesan cheese
If the butter hasn’t already begun to brown, increase the heat under the frying pan and cook butter until lightly browned. Pour the browned butter over the top of the eggs. Season with salt and freshly ground pepper. Serve immediately.