Prep the mussels: transfer the mussels to a larger colander and use a clean scrubbing brush or the rough scourer side of a new kitchen sponge to clean the mussels under running tape water, scrubbing off any remnants of seaweed or dirt, and removing the beards.
Cook the aromatics: in a large heavy pot over low heat, melt the butter, add the extra virgin olive oil, onion, garlic, celery, bouquet garni, and star anise, and stir to combine. Cook low and slow, stirring occasionally until soft and fragrant, around 6-8 minutes or so depending on the pot you use.
Create the sauce: Add the wine, stir, cover with the lid, increase the heat to high, and bring to a boil, to burn off the alcohol. Add half the cream, stir, cover, and cook until bubbling to reduce the sauce.
Steam the mussels: Again, working quickly, remove the lid, add the mussels, stir, and return the lid again, leaving the heat on high so the sauce can continue to reduce and the heat cook the mussels for 3 minutes or so or until they open.
Finish: remove the lid, add the remaining cream, sprinkle on half the parsley, and stir through.
Serve: Use a large serving spoon to transfer the mussels to bowls, pour the creamy sauce over the mussels, shower with more parsley, grind on some cracked black pepper, and serve with fries, mayonnaise, and crusty bread for soaking up the sauce.