Prepare your favorite French fries according to package or recipe directions.
De-beard the mussels by pulling their beard in an upward motion until you can pull it out of the shell. Clean their shells under running water to remove any barnacles. Discard any mussels that are open or won't close when you press on them.
Put the mussels in a bowl of salted water (1 tablespoon salt per quart of water) for 10 to 15 minutes. Use your hands to remove the mussels from the water and rinse them well in a colander. Don't pour the mussels into the colander to avoid any sand or grit getting back inside the mussels.
Heat the olive oil in a large, deep pot. Add the shallot and garlic, season lightly with salt and pepper, and cook over high heat, stirring, until the garlic is softened and lightly browned, about 3 minutes.
Add the mussels and cook, stirring, for 1 minute. Add the wine, cover, and steam the mussels until they open about 5 minutes. Remove from the heat.
Transfer the mussels to two deep bowls using a slotted spoon, and discard any mussels that did not open. Do not discard the remaining liquid from the pot.
Add the butter and parsley to the broth, swirling and shaking the pot until the butter melts.
Ladle the wine sauce over the mussels. Garnish with chopped parsley and serve toasty, crusty bread and pommes frites (French fries).