For Pesto Sauce:
Place basil leaves and garlic in bowl of food processor and pulse until well chopped.
Add flesh of avocado, olive oil, salt & pepper. Blend until smooth.
Add juice of lime and blend until well blended. Set aside. If not using immediately, place in well covered container and place in refrigerator.
Cook pasta according to package directions. Drain. Reserve about 1/2 cup pasta cooking water.
When pasta is cooking, slice cherry tomatoes and mozzarella balls into halves. Set aside.
Place drained pasta back into pan, add warm pasta cooking water and pesto sauce. Stir until well combined.
Stir in tomatoes and mozzarella balls.
Place pasta into large serving bowl and top with lots of freshly grated parmesan cheese.