Bring a small saucepan of water with a pinch of salt to a boil. Cook the eggs to your desired level of doneness: 3 minutes for soft-boiled, 5 minutes for medium-soft, or 7-8 minutes for hard-boiled.
Peel the asparagus and snap or cut off the tough ends.
Bring a large pot of water with a pinch of salt to a boil. Once the water is boiling, add the asparagus and cook for 5–7 minutes, depending on the thickness. Turn off the heat and let the asparagus finish cooking in the hot liquid for 10 more minutes.
Meanwhile, prepare the asparagus sauce. Melt the butter in a small saucepan. Pour the melted butter into a small pitcher with a spout.
Place the egg yolks and a splash of lemon juice in a tall cup that is large enough to fit your immersion blender. Insert the blender and turn it on.
Slowly pour the melted butter into the mixture in a thin stream while blending at a steady speed. The sauce will thicken. Continue blending until all the butter is incorporated and the sauce is smooth and even.
Remove the asparagus from the pot and let it drain briefly. Arrange the asparagus on a serving platter. Peel and cube the eggs, then distribute them over the asparagus along with the sauce.
Sprinkle some finely chopped fresh parsley on top, to taste. Serve immediately. Enjoy your meal!