Scrub the mussels clean, throwing away any that are cracked, are gaping and don’t close or have a strong smell (they should smell like a clean beach). Pull off any pieces of the protruding beard.
Sweat the shallots, thyme and garlic in the butter in a large pot. Add the wine and bring to a simmer over medium heat. Add the mussels, sprinkle with salt, cover and turn the heat to high.
When the steam starts to billow from the pot, turn the heat back down to medium. Continue to cook for 2 more minutes. Remove the lid from the pot, stir the mussels to evenly cook, replace the lid and continue to cook 3 more minutes. Remove the lid and check to see if all the mussels are open. If not, return the lid and cook for another 2 minutes. The shells should be just opened, do not overcook the mussels. Scoop the mussels out of the pot with a large slotted spoon or skimmer into a large bowl. Cover lightly with foil and reserve.
Carefully pour the mussel steaming liquid through a fine mesh strainer lined with cheesecloth into a small saucepan, making sure that any sandy or gritty particles are left behind. Pour the hot steaming liquid over the mussels to warm. Sprinkle with finely chopped parsley and serve.