Preheat the oven to 350 degrees.
Prepare food processor with steel knife or blade in center. Place the parsley in the bowl of the food processor. Secure the top of the processor, but leave the feed tube open. Run the food processor, dropping the garlic clove halves in every 2 seconds, for a total of 10 seconds. Use a spatula to scrape down the sides of the processor bowl. Pulse the parsley garlic mixture an additional 3-4 times, until both ingredients are very finely minced.
Add the parsley-garlic mixture to the large bowl of squash, season with the salt and pepper to taste, and mix well. Scrape the processor bowl with the spatula to remove as much parsley and garlic as possible, but do not wash the bowl.
Divide the squash-parsley mixture among the gratin dishes or place in casserole.
Place mozzarella cubes and goat cheese in bowl of processor. Pulse about 15 times, scraping down once or twice along the way, until the cheese is very finely mixed.
Place about 3 tablespoons of the cheese mixture on top of the squash, if using gratin dishes (if using casserole, sprinkle cheese mixture evenly over the top.)
Bake at 350 for 20 minutes, or until cheese is melted. Switch oven to broiler and broil until cheese is golden brown, watching carefully.
Remove from oven and allow to cool for 5 minutes. While the squash is cooling, chiffonade the basil leaves.
Garnish with the squash with basil and serve.