Bring a large pot of salted water to boil and add the pasta, cooking according to packet instructions or until al dente (typically 11-12 min for spaghettoni, depending on the brand).
While that’s happening, we make the sauce:
In a bowl add your eggs (whole and yolks) and beat well. Stir through the grated pecorino and the black pepper. Set aside.
In a separate pan, heat the EVOO and add the asparagus stems that have been cut into rounds – sauté on a medium heat until tender with a nice colour – a little char here is ok.
Add the spears of the asparagus to the same pan that you’re cooking the pasta a few minutes before the pasta is very al dente, then about a minute before the pasta is ready add all of that directly to the pan with the asparagus (reserving the pasta water) taking some of the pasta water with it. Turn the heat back on the lowest flame, and allow the pasta to continue to gently cook, giving it a toss in the oil that the asparagus cooked in.
Add roughly 1/4 cup of the pasta water to the egg mixture and stir well until the pecorino and egg mix is nice and smooth and quite liquid, like pouring cream.
Take the pan with the pasta off the heat, pour in the egg mix and stir, toss and mix continuously until it starts to thicken slightly and you have a stable emulsion. Add pasta water as needed, and equally if it’s too thin for your liking you can place the pan back over a very low flame – stirring constantly – to allow it to thicken.
Divide between two bowls, draping a couple of slices of prosciutto over each one to serve.