Step 1Clean the mussels: Inspect each mussel, discarding any that have opened (if some are partially opened, tap them with your finger and if they close back up they are OK) or have cracked shells. Wash the mussels, swishing them in a bowlful of water and scrubbing if necessary before thoroughly rinsing. Remove the beards no more than an hour before cooking. Set aside.
Step 2In a heavy, straight-sided, lidded frying pan or casserole, heat 1 tablespoon olive oil over medium heat and add the sausage. Cook, stirring until it begins to brown and release fat, then stir in the potatoes. Reduce the heat slightly, cover and cook until the potatoes are lightly browned and tender, stirring often, 10 to 15 minutes. Season with ½ teaspoon salt and several grinds of pepper, or to taste. Remove from heat and transfer the mixture to a bowl. Set aside.
Step 3Heat the remaining olive oil in the pan over medium heat. Add the onion and cook, stirring and scraping the bottom of the pan, until the onion begins to soften, about 3 minutes. Add a generous pinch of salt, cover, and reduce the heat to low. Continue to cook, stirring often, until the onion is tender and lightly colored, about 15 minutes. Add the garlic and thyme and cook, stirring, for another minute.
Step 4Stir the sausage and potatoes back into the pan with the onions. (At this point you can set the pan aside off the heat, until ready to serve. Reheat before continuing.) Add the wine and increase the heat to high, bringing the wine to a boil. Reduce the heat slightly and boil the liquid for two minutes. Add the mussels and cover the pan tightly. Cook 2 minutes, shake the pot vigorously and cook another 2 minutes. Transfer the opened mussels to soup bowls (if any haven’t opened, give them another minute or so to cook, then discard any unopened ones).
Step 5Taste the broth and adjust the seasoning if desired. Stir in the parsley (or parsley and cilantro). Using a slotted spoon, distribute the sausage, potatoes and onions equally among the bowls, then spoon over the broth and serve.