Snap the woody ends off the asparagus spears and discard. Heat a griddle pan until very hot and sprinkle it with the oil.
Meanwhile, bring a frying pan of water to the boil for the eggs.
Place the asparagus on the griddle and cook, turning regularly, until slightly charred at the ends.
Stir the boiling water vigorously then, one at a time, drop the cracked eggs into the centre – the swirling water will help the egg white collect around the yolk and maintain the shape of the egg. Cook for 4–5 minutes then remove with a slotted spoon.
Serve the asparagus on 4 warmed plates, and top with the poached eggs.
Using a vegetable peeler, make curly shavings of Parmesan and sprinkle on top of the asparagus.