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Pour the milk and water into a large based pan and add the bay leaf and black peppercorns. Bring the liquid to the boil then turn down to a simmer. Place the fish in the milk mixture and poach gently for 3-4 minutes. Remove the fish from the pan and set aside. Once cooled break the fillet into bite-sized flakes, ensuring there are no bones.
Heat the butter gently in a small grill-proof frying pan. Meanwhile beat the eggs together and season with salt and pepper. Add the eggs to the frying pan and stir gently to ensure the eggs start to set. When the bottom of the omelette is set but the top is still wet take the pan off the heat.
Place the flakes of fish over the omelette, sprinkle over the parmesan cheese, single cream and chopped parsley.
Place the pan under a medium-hot grill and cook the omelette on the top until just cooked and lightly browned.
Remove the pan from the grill and immediately serve the omelette onto a plate to stop it cooking. Enjoy straight away.