Melt the butter in a Dutch oven set over medium heat. Add the shallots and the garlic; cook until transparent, about 3 minutes. Add the thyme, bay leaves, and wine. Bring mixture to a boil over high heat.
Add mussels, and cover. Cook, shaking pan often, until mussels open, 5 to 6 minutes. Discard any mussels that fail to open. Using a slotted spoon, transfer mussels to a large bowl, reserving liquid in pan. Sprinkle 1/4 cup parsley over the mussels.
Remove and discard bay leaves and thyme. Return liquid to low heat; stir in crème fraîche or sour cream and remaining 1/4 cup parsley. Cook just until warm again, and pour over mussels. Serve.
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