Make the court-bouillon. In a shallow pan or a fish poacher, combine the water, wine, carrots, onions, thyme, and bay leaves. Squeeze the juice from the lemon into the pan and add the lemon shells. Add the salt and the black pepper and bring to a boil over medium-high heat. Reduce the heat to medium-low and simmer for 30 minutes. Remove and discard the lemon shells. Simmer for 12 minutes more.
Meanwhile, make the potatoes. Put the water and 1/2 teaspoon of the salt in a large saucepan and bring to a boil over high heat. Add the potatoes and reduce the heat to medium. Cook until fork-tender, about 25 minutes. Drain.
Season the fillets with the salt. Add the fillets to the court-bouillon, cover, and cook until the fish flakes easily with a fork, about 10 minutes.
Meanwhile, make the lemon butter sauce. In a small nonreactive saucepan, combine the shallots, lemons, wine, salt, and cayenne over medium heat, bring to a gentle boil, and cook until as thick as syrup. Add the cream and cook, stirring a few times for 2 minutes, then remove from the heat. Add the butter 1 tablespoon at a time, whisking after each addition, until all is incorporated. Add the hot sauce and Worcestershire and whisk to blend. Strain through a fine-mesh sieve and keep hot.
Finish the potatoes. In a medium-size skillet over medium heat, melt the butter. Add the potatoes, parsley, the remaining 1/4 teaspoon salt, and the white pepper and cook until heated through, 2 to 3 minutes.
To serve, put a fillet on each serving plate, spoon 1/4 cup of the sauce over each, and serve with equal portions of the potatoes.
Yield: 4 servings