In a large pot, sweat the new onions in 50 g (3 tablespoons) butter, add the cultivated mussels and the thyme branch and combine; add the dry white wine. Cook covered, stirring often until the mussels open. Drain the mussels, reserving the cooking liquid as well as the new onions.
Remove half of each mussel shell and place the half containing the mussel on a large plate, forming several concentric circles. Place on the oven door to keep warm.
Pour the mussel liquid with the onions into a saucepan, remove the thyme branch, and incorporate the remaining butter a little at a time, whisking constantly over low heat. Add the chopped herbs.
Pour the sauce over the mussels, add a grinding of fresh pepper and serve immediately.