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1
Combine the 00 flour, instant yeast and salt in a large bowl. Add the cold water all at once and stir vigorously with a wooden spoon until a very slack, cohesive batter forms — it will be wetter than any workable dough, closer to a thick pancake batter. Cover the bowl tightly and refrigerate for 12–18 hours. The dough is ready when it has at least doubled in volume and the surface is domed and studded with bubbles.
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2
Pour the olive oil into a deep, heavy pan to a depth of at least 6 cm and set over a medium-high flame. Bring the oil to exactly 180 °C, verified with a thermometer. This is where the dish lives or dies: oil below this temperature produces pettole that absorb fat and emerge heavy and greasy; oil above it browns the shell before the interior cooks through. Maintain the temperature between drops throughout frying.
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3
Wet both hands thoroughly. Using a wet dessertspoon, drop small portions of the fermented batter — no larger than a walnut — into the hot oil in batches of five or six, never crowding the pan. The pettole will self-roll as they fry; leave them undisturbed for the first minute, then turn once. Fry for 3–4 minutes total until each pettola is evenly golden and puffed. Remove with a slotted spoon and drain briefly on a rack. The pettole must be served within minutes; they deflate and toughen as they cool.
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4
While the first batch fries, begin the anchovy sauce. Warm the extra-virgin olive oil and garlic together in a small saucepan over the lowest possible heat until the oil reaches approximately 50 °C — it should feel hot to a cautious fingertip but must not shimmer or smoke. Add the desalted anchovy fillets and stir gently; they will dissolve into the oil within 3–4 minutes. The sauce is ready when no solid anchovy remains visible and the oil has turned a rich amber.
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5
Remove the garlic. Add the rinsed capers and the crumbled dried chilli to the anchovy oil. Stir once and remove from the heat. Taste: the sauce should be assertively saline, rounded with the gentle heat of the chilli. Transfer to a warm serving vessel and bring to the table alongside the hot pettole.