Directions Cut the fish fillets into about 7 1/2cm/3in pieces.
Put the trimmings and heads into a large saucepan with all the remaining soup ingredients, except the bread and aioli, and bring to the boil.
Half cover the pan and simmer for 30 minutes.
Meanwhile, make the aioli.
Soak the breadcrumbs in vinegar for 5 minutes, then squeeze dry.
Transfer the crumbs to a bowl and mash in the garlic.
Beat in the egg yolks, one at a time, then the salt, mustard and pepper.
When the mixture is very thick and smooth, beat in the olive oil, a few drops at a time.
Do not add the oil too quickly or the mayonnaise will curdle.
When the mayonnaise has thickened, the oil may be added a little more rapidly.
When all the oil has been added, beat in the lemon juice.
Set aside.
Strain the fish stock, pressing down on the vegetables with the back of a wooden spoon.
Rinse out the pan, then return the fish stock to it.
Add the fish and bring to the boil.
Reduce the heat to low and simmer for 10 minutes, or until the flesh flakes easily.
Using a slotted spoon, transfer the fish pieces to a warmed serving bowl.
Boil the stock rapidly for 10 minutes or until it has reduced a little.
Very gradually beat about 125ml/4floz (1/2 cup) of the fish stock into the aioli.
Pour the mixture into the saucepan and simmer gently for 1 to 2 minutes, being careful not to let it curdle.
Arrange the toasted bread rounds on the bottom of individual bowls, and cover with fish pieces.
Pour over the aioli stock and serve at once.
