Bring a large pot of salted water to a boil over high heat. Add the asparagus bundle and cook for 3 minutes Use tongs to transfer the asparagus to a bowl of ice water and let sit until cool, about 5 minutes. Drain the asparagus, remove the rubber band, and cut into 1-inch pieces.
Meanwhile, add the pasta to the same 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 the pasta and place in a large bowl. Add the basil aioli, asparagus, shaved cheese, salt, and pepper. Stir to combine and turn out onto a serving platter.
Melt the butter in a medium skillet over medium-high heat. Add the eggs, sprinkle with salt and pepper, fry 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 stored, covered, in the refrigerator for up to 3 days.
There are some dangers associated with eating raw eggs, especially for those with compromised immune systems. To reduce the risk of salmonella or other food-borne illness, use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells