Cut the cooked asparagus spears into thirds. Put in a food processor with the basil, pine nuts, garlic, and salt and pepper, to taste. Keep in mind you will add Parmesan as well, so be careful not to oversalt. With the machine running, slowly add the cup of olive oil. When the sauce is about the consistency of mayonnaise, it has enough oil. Pulse in the Parmesan. Thin with water if necessary to achieve a slick, saucy pesto. Scrape into a bowl or jar, cover, and refrigerate until needed. You should have about 3 cups. (Keeps about 2 to 3 days, refrigerated.)
Put the potatoes in a pan of salted cold water to cover and bring to a boil. Cook until tender, about 10 minutes. Drain and let cool for a few minutes. If the potatoes are larger than about 1/2 inch in diameter, cut in half.
Heat the olive oil in a medium saute pan over medium-high heat until hot. Add the potatoes and cook until browned and crispy all over, about 5 minutes. Season, to taste, with salt and pepper. Keep warm. Add basil, toss with potatoes and set aside.
While the potatoes are browning, return the water used for cooking the asparagus, if reserved, to a boil or brint a new pot of water to a boil. Add more salt and the pasta and cook until al dente, about 12 minutes. Drain through a colander, reserving about 1/2 cup of the pasta cooking water. Pour the pasta into a warm serving bowl, add the potatoes, about 1 cup of the asparagus pesto, several twists of pepper, and 1/4 cup of the Parmesan. Toss well, thinning with the pasta cooking water as necessary.
Dust with a light grating of Parmesan (a microplane grater works well for this) before serving and pass the remaining cheese and a bowl of pesto at the table.