Discard any mussels that are cracked or open and will not close with a tap on the counter. Place the rest in a large container and cover with cold water for 20 minutes to filter out any sand. Lift the mussels from the water, leaving all the sand behind. Remove the beards by grasping them with a dry towel and pulling toward the hinges of the shells. Scrub the mussels under cold running water.
Heat 1 tablespoon of the butter in a large saucepan over medium heat. Add half of the onions, thyme, garlic, and bay leaves and cook, stirring, for 3 minutes. Add 2 cups of the stock and the wine and bring to a boil. Toss in the mussels, cover, and simmer until the mussels open. Drain the mussels through a cheesecloth-lined colander set over a bowl and reserve the liquid for the soup. Pick the mussels from the shells, place in a shallow container, and cover with enough cooking liquid to keep them moist. (At this point the mussels and cooking liquid can be reserved, chilled, overnight.)
In a large Dutch oven or heavy-bottomed pot, melt the remaining 2 tablespoons butter over medium heat. Add the remaining onion, thyme, garlic, and bay leaf and the saffron threads and cook, stirring, until the onions are translucent. Add the cauliflower and cook, stirring, for another 5 minutes. Add the reserved cooking liquid and the remaining chicken stock. Simmer, stirring occasionally, for 30 minutes, or until the cauliflower is very tender. Add the heavy cream and crème fraîche and bring to a simmer. Discard the bay leaf and thyme and transfer the mixture to a blender (you will need to do this in batches). Puree until smooth and season with salt and pepper. Return to the pan, bring to a simmer, and stir in the mussels just before serving.
Serve, garnishing with the toasted bread and chives.