Put the haddock in a pan with the shallots and just cover with water. Bring to the boil and simmer, covered with a lid, for 2 minutes. Remove the haddock with a slotted spoon and drain on some kitchen paper.
Continue to simmer the cooking liquid until it has almost completely reduced, then add the double cream, bring back to the boil and continue to simmer until it has reduced by two-thirds and has thickened. Remove from the heat and leave to cool a little.
Flake the haddock flesh into the sauce, add the egg yolk and parsley, stir well and season with salt and freshly ground white pepper. Preheat the grill to its maximum temperature.
Heat a little butter over a low heat in a small non-stick egg or blini pan or a favourite frying pan that doesn’t stick. Season the beaten eggs and pour a quarter of the egg mix into the pan, or enough to fill two-thirds of the pan. Over a low heat, stir the eggs with a wooden spoon or plastic spatula, until the mixture begins to set but the eggs are still soft. Stop stirring.
Take a plate a little larger than the pan and turn it upside down on top of the pan. Invert the omelette on to the plate and transfer to a heatproof plate or serving dish. Repeat with the rest of the mixture until you have 4 small omelettes. Spoon the haddock mixture over the omelettes, spreading it evenly with the back of the spoon until covered. Put them under the grill for a minute or two until evenly browned. (If you left them in the pan to glaze, the insides of the omelettes would overcook and dry up in the pan’s residual heat.)