Bring a large pot of salted water to a boil over high heat. Add the asparagus and cook for 3 minutes. Transfer the asparagus to a bowl of ice water and let sit until cool, about 5 minutes. Remove from the water, remove the rubber band, and cut into 1-inch pieces.
Meanwhile, add the pasta to the large pot of salted water used to cook the asparagus and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain pasta into a large bowl. Add the Basil Aioli, cut asparagus, shaved cheese, salt, and pepper. Stir to combine and place on a long, rectangular serving dish.
Place the butter in a medium skillet over medium-high heat. Add the eggs and sprinkle with salt and pepper. Fry the eggs to your liking. Place the eggs on top of the pasta and serve immediately.
Combine the garlic, egg yolks, mustard, lemon juice, basil, salt, pepper, and cayenne in a food processor and run the machine to mix. With the machine running slowly drizzle in the vegetable and olive oil. Basil Aioli can be kept, covered, in the refrigerator for up to 3 days.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.