Poach the Haddock: In a pan, combine milk, peppercorns, chopped onion, tarragon, and haddock. Cover and poach for around 5 minutes.
Make the Sauce: Melt some butter in a separate pan, stir in the flour and cook for a few minutes. Gradually stir in some of the milk used for poaching the haddock, continuously stirring. Add mustard, Worcestershire sauce, and season with salt and pepper. Once smooth, add grated cheese off the heat and mix to emulsify.
Prepare the Omelette: Beat the eggs and season with salt and pepper. Cook with butter in a non-stick pan on low heat, pulling the sides into the middle as it cooks. Transfer to a cast iron skillet to stop cooking.
Assemble and Grill: Remove the skin from the haddock, flake it over the omelette. Mix egg yolks into the sauce and spoon over the haddock. Top with more grated cheese and grill until golden. Optionally, finish with a blowtorch for extra char.
Garnish and Serve: Top with freshly cut chives before serving.