Cook the potatoes in boiling water until they are tender. Meanwhile preheat the oven to 200 C, 180 C Fan Gas 6.
When cooked, slice the potatoes into a baking dish. Add the asparagus tips, sprinkle with salt and black pepper and drizzle with the olive oil. Turn the vegetables until they are evenly coated.
Bake until the asparagus is tender, about 15-20 minutes depending on how thick the asparagus tips are.
Fry the eggs in the butter in a large frying pan, sprinkle with salt and serve on top of the vegetables. The yolks should still be runny.