Cut off the asparagus tips, put them in a bowl, and cover with plastic wrap.
Cut the stems into 1-inch pieces and puree them in a blender, adding just enough of the reserved cooking water to get the puree going. Transfer the puree to a small saucepan. Bring to a simmer, then swirl in the butter, keeping it moving till the butter is completely incorporated. Add a four-finger pinch of salt. Keep warm.
Fry the eggs over high heat till the white is set, or cook as you wish.
While the eggs are cooking, reheat the asparagus tips in a microwave (you might want to toss with a little olive oil or butter first).
Toss however much pasta you feel like serving with the sauce till it's evenly coated. Divide among plates. Grate cheese over each and top with the asparagus tips, followed by the egg.
Crack pepper over the eggs and serve.