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.
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.