Bring a large pot of salted water to a boil and add asparagus. Cook for 3 minutes, then remove with a slotted spoon and place in a bowl of ice water. Let cool, then drain.
Bring the same pot of water back up to a boil. Add pasta and cook for 11 minutes, or until al dente. Then drain, reserving about ¼ cup of cooking water.
In a large bowl, combine pasta, asparagus, and basil pesto. Toss until pasta is evenly coated with pesto. If the sauce is too thick, begin pouring in some reserved pasta water (about 2 tablespoons at a time) until the sauce reaches desired consistency. Add chives, lemon zest, and mozzarella, and stir until just combined.
Divide pasta into bowls and top with Parmesan cheese. If eating immediately, serve hot. You can also serve it cold or at room temperature as a delicious vegetarian pasta salad.
