Rinse mussels in cold water to clean off any dirt or debris. By tugging in the direction of the shell's hinge, remove the "beard."
A dead or fractured mussel should be discarded. A mussel's shell can be tapped to determine whether it is dead. It is excellent and still living if it shuts. Throw it away if it doesn't close. Fresh mussels have to have an oceanic scent and be glossy, wet, and black.
After cleaning the mussels, have a big stock pot or Dutch oven ready. Over medium-high heat, preheat the olive oil in the saucepan. Add the shallots and garlic. For one minute, stir.
Stir the mussels with the oil and garlic-shallot purée after adding them. Add the butter and whisk just enough to melt it. Stir in the wine just a little. Add the parsley, heavy cream, and salt to taste (approximately two generous pinches). To combine all components, stir.
Simmer for 10-15 minutes, stirring once halfway through. Cover and lower the heat. Serve mussels right away with crisp French fries or a fresh toasted baguette (for a Moules Frites recipe). French fries and garlic toast with additional fresh chopped parsley as garnish.