In a small bowl, whisk 3 tablespoons of the oil with the lemon juice and mustard. Season with salt and pepper.
In a large bowl, toss the asparagus with the remaining oil and sprinkle with kosher salt. On a sheet pan lined with foil, roast the asparagus at 450 degrees until it's tender, 8 to 10 minutes, and keep it warm.
In a large skillet, melt the butter over medium heat. Cook the eggs sunny-side up until the whites are firm and the yolks are runny, about 5 minutes.
In a large bowl, toss the greens with the vinaigrette and transfer the salad to plates. Top with asparagus, fried eggs, and and cheese and serve immediately.