How to Make It
In a small bowl, whisk 3 tablespoons of the olive oil with the lemon juice and mustard. Season with salt and pepper.
Light a grill or preheat a grill pan. In a large bowl, toss the asparagus with the remaining 1 tablespoon of olive oil and season with salt and pepper. Grill the asparagus over high heat until tender and charred in spots, about 5 minutes; keep warm.
Meanwhile, in a large nonstick skillet, melt the butter over moderate heat. Working in batches if necessary, cook the eggs sunny-side up until the whites are firm and the yolks runny, about 5 minutes. Season with salt and pepper.
In a bowl, toss the arugula with the lemon vinaigrette and transfer the salad to plates. Top with the asparagus, fried eggs and cheese shavings and serve at once.