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1
Warm the olive oil in a marmite or heavy-bottomed pot over a medium flame. Add the onions, garlic, dried guindilla, and chopped piparrak. Cook gently, stirring occasionally, until the onions are entirely soft and translucent and the peppers have begun to collapse — the sofregit should smell sweet and deeply aromatic, with no raw edge remaining.
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2
Add the tomato concassée and stir through. Allow it to reduce into the sofregit until the liquid from the tomatoes has evaporated and the mixture darkens slightly at the edges of the pot.
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3
Add the potato chunks in their irregular cuts. Pour over sufficient water or light fish fumet to cover. Bring to a steady simmer and cook for 20–25 minutes. The potatoes are done when they are tender and their edges have begun to crumble, releasing their starch into the broth; the liquid should thicken perceptibly and coat the back of a spoon.
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4
This is where the dish lives or dies. Nestle the tuna chunks among the potatoes and cover the pot. Cook for 5–6 minutes only, over a gentle simmer — never a rolling boil. Remove a piece and press it lightly: it should yield with a slight resistance at the centre. Cut it open; the interior must remain rose. An internal temperature of 55–58 °C confirms it. Tuna taken beyond this point will be dry and mealy and the dish cannot be recovered.
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5
Remove from the heat. Add the piment d'Espelette generously and scatter the flat-leaf parsley across the surface. Taste and adjust salt. The stew should be thick and chunky — more stew than soup — with visible pieces of tuna among broken potatoes and peppers. Ladle into deep bowls, drizzle with Basque olive oil at the table, and serve immediately with crusty bread.