How to Make It
Preheat the oven to 400°. Bake the potatoes until tender, about 40 minutes.
Shortly before the potatoes are done, set a large cast-iron skillet over high heat. When it starts to smoke, add the mussels and sprinkle with the thyme. Cover and cook, shaking the pan occasionally, until all the mussels open, about 4 minutes. Transfer the mussels to a platter and cover with foil to keep warm; discard any that do not open.
Boil the mussel juices in the skillet for 2 minutes to concentrate the flavor, then whisk in 4 tablespoons of the butter to make a rich pan sauce. Melt the remaining 4 tablespoons of butter in a small saucepan and add the lemon juice.
Split the potatoes and pinch them slightly to make the insides puff up. Pour the pan sauce into the potatoes and sprinkle with coarse salt. Serve surrounded by the mussels, with the lemon butter alongside for dipping.