Preheat oven to 400 degrees F.
Meanwhile fill a large fry pan or sauté pan with 1 inch of water, add the salt and bring the water to a rolling boil. Carefully add the asparagus to the pan and cook uncovered over medium heat, just until tender, about 8-10 minutes, depending on the thickness of the stems.
Grease a large baking pan liberally with olive oil. Carefully place the drained asparagus, in a single layer, on the pan. Sprinkle the cheese liberally over the asparagus and dot them with the butter. Place the pan into the oven and bake for 10 minutes or until the cheese has melted and is beginning to brown. While the asparagus are in the oven, fry the eggs. Transfer the asparagus onto a serving dish separating them into 4 servings. Top each serving with a fried egg. Serve immediately.
White asparagus can be substituted for green if desired. The asparagus can also be cooked in a tall pot such as an asparagus cooker.