Preheat the broiler.
Bring a small pot of water to a boil.
Prepare a small bowl of water with ice in it.
Trim the tough ends off the asparagus. Place the spears on a small baking sheet or dish, drizzle with olive oil, and season with salt and pepper. Cut the lemon in half and then into small thin slices, and sprinkle them over the asparagus.
Place the asparagus under the broiler.
Place the eggs gently in the boiling water to cook for 6 minutes for a very soft egg, 7-8 minutes for a slightly firmer yolk, and 10 minutes for a set yolk. Remove them from the pan and place in the ice water for a couple of minutes.
Keep an eye on the asparagus – if your broiler is very hot, it may be done. The tips should be nice and crispy and the lemon slices caramelized. When done, remove from the pan to a warm platter. Make sure to pour all the juices from the pan over the asparagus and drizzle with a little extra olive oil
Peel the eggs and cut them in half. Arrange alongside the broiled asparagus. Top with optional ingredients, if using, and serve.