Place the fish heads, bones and trimmings into a large heavy pot. Add the chopped onions and pour over the cold water. This slow extraction forms the backbone of the soup, so use only cold water and begin gently before moving to the next step.

Bring the pot to a steady boil over medium heat, then reduce slightly and simmer uncovered for 40 minutes. Stir occasionally, allowing the onions to soften completely and dissolve into the liquid. This creates the naturally thick texture Halászlé is known for.

Remove the pot from the heat and strain the broth through a fine sieve, pressing firmly on the solids to extract maximum flavour. Discard the solids and return the strained liquid to the pot, ready for seasoning.

Stir in the sweet paprika and hot paprika while the liquid is warm but not boiling. This protects the paprika from turning bitter and preserves its colour. Mix thoroughly before returning the pot to the heat.

Add the sliced green pepper, chopped tomato, and salt. Bring the soup back to a gentle boil, allowing the flavours to merge for 10 minutes before adding the fish pieces.

Carefully add the fish chunks to the boiling soup. Do not stir, instead gently shake the pot to prevent breaking the fish. Cook for 10 to 12 minutes until the fish is just tender and fully cooked.

Taste the broth and adjust salt or paprika if needed. The soup should be bold and warming, with a clean fish flavour rather than heaviness. Allow it to simmer for 2 minutes more before serving.

Serve the Halászlé piping hot in deep bowls with thick slices of fresh bread. Traditionally it is enjoyed without garnish, allowing the colour and aroma to speak for themselves. A small extra spoon of hot paprika may be offered at the table for those who prefer more heat.