How to Make It
In a large saucepan, cover the potatoes with water and add a large pinch of salt, bring to a boil and simmer over moderately high heat until the potatoes are tender, about 20 minutes. Drain and let cool, then slice the potatoes 1/4 inch thick and transfer to a bowl.
Heat the olive oil in a large pot. Add the garlic, thyme sprigs and onion and cook over moderately low heat until softened, about 12 minutes. Add the rosé and 2 tablespoons of the lemon juice and bring to a boil over high heat. Add the mussels, cover and cook, shaking the pan a few times, until the mussels open, about 7 minutes. Set a colander over a large bowl. Pour the mussels into the colander and when cool enough to handle, shell them. Add the mussels to the potatoes and cover with plastic wrap.
Slowly pour the mussel juices back into the pot, stopping before you reach the grit at the bottom. Bring to a boil over high heat and cook until reduced to 1/3 cup, about 12 minutes. Let cool.
In a medium bowl, whisk the mayonnaise with the remaining 3 tablespoons of lemon juice and the mustard. Stir in the capers, cornichons, parsley, chives and the reduced mussel juices. Season with salt and pepper. Fold into the potatoes and mussels. Serve the salad at room temperature or slightly chilled.