Begin by preparing a batch of fresh egg pasta, then a good portion of béchamel sauce (see Notes for links to the recipes).
Now prepare the asparagus: Trim and peel the asparagus, then boil, or better, steam it in lightly salted water until tender—not crisp tender or mushy, but fully tender. Cut off the tips and set them aside, and purée the remaining stems in a blender or food processor with a bit of béchamel sauce. Season the purée with salt to taste—it should be quite savory.
Now assemble your lasagna just as you would classic lasagne: Parboil the pasta sheets in well salted water for just a minute or two. Drain and pat dry. Place a layer of pasta in a buttered baking dish, then a thin layer of béchamel, followed by a layer of asparagus purée, then place a few asparagus tips here and there, then sprinkle generously with grated parmesan cheese.
Keep on layering in this way until you have used up your ingredients—or until you have laid down four or five layers of pasta. (This kind of lasagna should not be too thick.) Cover with a layer of béchamel, sprinkle with more grated parmesan and dot with butter.
Bake in a hot oven (400F/200C) for about 20 minutes or so, until the top is lightly browned. Allow the dish to settle and cool for 10-15 minutes or more—see below—before serving.