Depending on what kind of flatfish you're using, either remove the top dark skin by pulling it off or cutting it off with a sharp or - if the skin is relatively thin- leave it attached or scale it. Scrape the bottom white side to remove any scales and remove the sharp fins with scissors. (Or ask your fishmonger to do all of this.) Rinse the fish in cold water.
Preheat the oven to 400 degrees. In a 4-quart pot, bring the wine to a simmer with the shallots. Add the mussels or clams, cover, and steam until they open - about 7 minutes for mussels, 12 minutes for clams. Take the mussels or clams out of their shells, discard the shells, and reserve the meats. If the shellfish-steaming liquid is sandy, pour it slowly and carefully into a bowl, leaving the sand behind. Arrange the flatfish- dark skin side up- in a buttered or oiled dish that closely fits it and pour on the shellfish- cooking liquid. There should be enough liquid to come halfway up the sides of the fish. If there isn't, add some water or fish broth. Place the containers with the fish on the stove over medium heat until the liquid comes to a simmer. Loosely cover with aluminum foil and slide the fish into the oven . Braise for 5 to 12 minutes per inch of thickness. Carefully transfer the fish to individual platters or plates using a large spatula. Keep the fish warm while you're preparing the sauce. Strain the cooking liquid into a 2- quart saucepan and cook it down over high heat until 1/2 cup remains. Thicken the sauce by whisking in any optional thickeners and bringing it to a quick simmer. Once the sauce is thickened, add the lemon juice and parsley and season with salt and pepper to taste. Gently reheat the mussels or clams in the sauce - don't let it boil - and spoon the sauce and the mussels or clams over each fish.