Scrub and de-beard mussels in running cold water. Discard mussels with cracked shells.
Melt butter in a large stock pot over medium heat. Add the shallots and 1/2 of the chopped garlic and sauté for about 3 minutes, stir often so garlic does not burn.
Add wine, turn heat up to high and bring to a boil for 3 minutes.
Add mussels and cover. Cook for about 7 minutes until mussels are open. Scoop out mussels into a bowl. Cut heat to medium, pour in 1/2 cup cream, cook for 2 minutes while stirring.
While stock is cooking, in a blender or food processor, puree the basil, parmesan, olive oil, 3/4 cup of cream, and remaining garlic. If you have difficulty with the basil puree add some of the stock to your blender.
Add the basil puree to the stock, season with salt and pepper and simmer for 3 minutes.
Pour over mussels, garnish with parsley, and serve hot.
As an alternative method, you can skip the blender instruction and simply add the basil puree ingredients to your stock pot and puree with an emersion blender. The recipe will yield a chunky sauce, the emersion blender will make the sauce more of a soup-like consistency.