Make the tartare sauce: mix all the ingredients together and chill (you won’t need it all; leftovers keep for 3-4 days, covered, in the fridge and enliven any seafood dish).
Preheat the oven to 150C/gas 2/fan 130C. Spread the crumbs over a baking sheet and dry in the oven for 10-15 minutes until pale golden, stirring halfway.
At the same time, hard-boil one of the eggs. Cut the potatoes into chunks. Cook in boiling salted water for 10-15 minutes until tender. Drain well and mash with a fork.
While the egg and potatoes are cooking, put the fish in a frying pan with just enough water to cover. Bring to the boil, reduce the heat, cover and simmer for 5 minutes until just cooked. Remove the fish with a palette knife and put on a plate. When it is cool enough to handle, carefully remove the skin and any bones. Flake the fish.
Shell the egg and chop it. Carefully mix with the potato, fish, lemon zest, parsley, ½ tsp salt and a some black pepper. With floured hands, shape into 4 round cakes.
Coat each cake in the egg, then the dried breadcrumbs. Reshape if necessary.
Heat the oil in a frying pan and fry the cakes over a medium heat for 5 minutes on each side, turning once, until golden on both sides. Serve with the tartare sauce.