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1
Make the fish fumet first. Place the reserved heads and bones of the monkfish, conger eel, and hake in a pot with cold water to cover, bring to a gentle simmer, and cook for 20 minutes, skimming throughout. Strain and reserve. The fumet should be clear and pale gold.
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2
Warm the olive oil in a wide, heavy pot over a medium-low flame. Add the onions, garlic, sweet long peppers, and piment d'Espelette. Cook, stirring regularly, for 30 minutes. The sofregit is ready when it is completely soft, fragrant, and beginning to colour at the edges.
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3
Add the peeled, chopped tomatoes to the sofregit. Continue cooking over medium heat, pressing the tomatoes down as they collapse, until the entire mixture thickens to a near-paste with no standing liquid — this is the point on which the stew lives or dies: a loose base will not give the broth its body or its brick-red depth.
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4
Pour in the fish fumet and the white wine. Stir to lift the paste from the base of the pot and bring to a steady simmer. Taste and adjust salt.
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5
Add the monkfish and conger eel to the simmering broth. The flesh should be submerged. Cook for 8 minutes; the exterior of the monkfish will be opaque and just beginning to yield at the thickest point when pressed.
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6
Add the hake pieces. Cook for a further 8 minutes. Hake is done when the flesh is white throughout and flakes at a gentle prod — it should not be pressed beyond this point.
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7
Add the langoustines, mussels, and squid. Cook for 5 minutes. The mussels must be fully open and the langoustines uniformly pink; discard any mussel that remains closed. Add a splash of Txakoli at this point if available, to restore brightness.
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8
Stir together the mayonnaise, piment d'Espelette, crushed garlic, and a drop of cooled fumet to make the rouille. Spread onto the garlic-rubbed croûtons. The rouille should be a deep terracotta colour and carry a pronounced fruity heat on the tongue.
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9
Ladle the stew into deep bowls, arranging the fish and shellfish to show their variety. Pour the broth generously around them — broth, fish, and shellfish unified in every bowl. Float two rouille-spread croûtons on top of each serving.