Prepare the asparagus: Trim the woody ends. Boil or steam the asparagus for 3–4 minutes until tender but still crisp. Drain and set aside.
Cook the eggs: For poached eggs: Bring water to a simmer, add a splash of vinegar, crack each egg into a small cup, and gently slide it into the water. Cook for 3 minutes and remove with a slotted spoon.
For fried or soft-boiled eggs: Cook to your liking.
Toast the croutons: Warm 1 tablespoon of olive oil and 1 tablespoon of butter in a skillet. Add the croutons and cook, tossing them, until they turn golden and crisp, about 2–3 minutes. Remove from the pan and set aside.
Assemble the dish: Divide the asparagus onto 4 plates. Top each serving with one egg. Sprinkle croutons over the top. Drizzle with a little olive oil. Season with salt and pepper. Add Parmesan or herbs if you like.
Serve immediately while warm.