In a medium pot, bring water to a boil and cook the pasta according to package. Reserve 2-4 tablespoons of pasta water before draining.
While pasta cooks, remove seed from avocado and scoop into a medium size bowl. Mash the avocado with a spoon or fork until smooth and creamy. Add pesto, lime juice, and a few cracks of salt and pepper and mix. For smoother consistency, see notes.
Drain your pasta, reserving about 4 tablespoons of pasta water. Add the pasta back to the pot with pesto mix, cherry tomatoes, mozzarella, arugula, and shredded chicken. Mix together, adding pasta water until it reaches a desired consistency.
Season with a few more cracks of salt and pepper and a pinch of shredded parmesan.