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Boil the vegetables to prepare a stock. Add the onion, celery, carrots, parsley, and potatoes to a pot with enough water to cover the vegetables. Boil for 30-35 minutes or until all the vegetables are cooked. Remove the boiled vegetables from the stock and leave it aside so you can poach the fish later.
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Meanwhile, prepare the mayonnaise. Mix the egg yolk and olive oil in a medium bowl, whisking all the while vigorously. Add mustard powder, keep whisking, and add the water, lemon juice, salt, and red wine vinegar. IMPORTANT: Each ingredient needs to be added very slowly, while whisking so that it is incorporated into the mix without curdling it. Whisk in the flour last to thicken the mayo.
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Carefully place the whole, cleaned fish in the pot with the vegetable and simmer it gently to poach for 15-20 minutes, depending on the size of the fish. Note that a whole trout weighing 10-12 oz (275-350 g) each will take 8-10 minutes – less for small fish.
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As soon as it’s ready, using two large slotted spoons, remove the fish very carefully and place it on a platter or cutting board. Let it cool a little, then clean the fish thoroughly, but keeping the head and tail intact. As you clean and shred the flesh, try to keep the fish flesh in big pieces.
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Place the fish in a mixing bowl, add capers and 2 tablespoons of mayonnaise, or more, as needed, to make a moist mixture that holds together.
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Reconstruct the fish on a platter, placing the head and tail at either end and use the boiled vegetables as a garnish around the fish. Top the fish with the rest of the homemade mayonnaise and serve.