Combine flour, salt and pepper in a shallow dish. Dredge fish in flour. Heat 2 teaspoons of the oil in a Dutch oven or large saute pan. Swirl the pan to coat the bottom evenly with oil, then add the fish. Cook, turning, just until lightly browned, 2 to 3 minutes. Transfer the fish to a plate and set aside.

Add the remaining 1 teaspoon oil to the pan and reduce heat to medium. Add onion and garlic and cook until softened, about 2 minutes. Slowly pour in wine and bring to a boil. Add broth (or diluted clam juice); bring to a boil. Add potatoes and carrots and return to a boil. Cover and cook over medium heat until the vegetables are just tender, about 20 minutes.

Return the fish to the pan, reduce heat to low and cook, covered, until the fish is opaque in the center, 5 to 10 minutes longer. Transfer the fish and vegetables to a plate with a slotted spoon and keep warm.

Increase the heat to high and boil the cooking liquid for 5 minutes to intensify the flavors. Whisk together egg, 3 tablespoons lemon juice and dill (or fennel) in a medium bowl. Gradually whisk a little of the hot cooking liquid into the egg mixture, then pour the egg mixture into the remaining cooking liquid in the pan. Cook over medium heat, stirring constantly, until slightly thickened (do not boil) and the liquid reaches a temperature of 160F on an instant-read thermometer, 1 to 2 minutes. Return the fish and vegetables to the pan and heat through. Taste and adjust seasonings, adding more lemon juice, if desired.