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1
Drain the soaked chickpeas and place them in a deep saucepan, cover generously with cold water, then add the rosemary sprigs, garlic cloves, and olive oil — starting in cold water and cooking the chickpeas slowly with oil in the pot encourages them to soften evenly from the outside in, so you avoid a skin that splits before the centre is tender. Bring slowly to a simmer and cook, uncovered, for 2 hours, replenishing with hot water as needed to keep the chickpeas submerged; adding hot water rather than cold keeps the simmer steady, so the chickpeas do not tighten and turn grainy. They are done when they crush completely between two fingers with no chalky centre.
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2
Reserve the cooking liquid, which carries the flavour of the rosemary and garlic and will thin the purée later without diluting it. Remove and discard the rosemary, then transfer the chickpeas and garlic to a blender and blend on high until uniformly smooth, adding a ladleful of cooking liquid to assist the motor. Pass the entire batch through a fine-meshed sieve, pressing firmly with a ladle — this second step catches every fragment of skin that the blender left behind, because those fragments would otherwise show as small bitter specks on the tongue; the purée is ready when it is perfectly smooth on the tongue with no skin particles, no grit.
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3
Return the sieved purée to a clean saucepan over low heat and add the reserved cooking liquid a little at a time, stirring constantly, until the purée flows off a spoon in a slow, continuous ribbon — adding the liquid gradually gives you control over the final consistency, since chickpea purée tightens noticeably as it cools and will seem too loose in the pan but correct in the bowl. Season with sea salt, adding more than seems necessary and tasting repeatedly, because chickpea purée absorbs salt slowly and will taste flat at the table if seasoned only once. Stir in a thread of olive oil and hold warm.
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4
Set a wide frying pan over high heat, add the butter, and let it foam and subside — waiting for the foam to subside tells you the water in the butter has driven off and the pan is truly hot, so the gamberi will colour rather than steam. Add the gamberi in a single layer and cook for 1 minute without moving them, then pour in the white wine and cook for a further 1 minute only, removing the pan from the heat the instant the gamberi are just opaque throughout and still yielding to the press of a fingertip, because even thirty seconds more will turn them firm and dry. Any residual heat in the pan will finish them; they do not need another moment on the flame.
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5
Warm four shallow bowls so the purée stays fluid from the first spoonful to the last, then ladle the chickpea purée into each, spreading it to a shallow pool. Arrange the gamberi on top and finish each portion with a thread of raw Tuscan olive oil, a tiny pinch of smoked salt, and exactly 3 drops of aged balsamic — the balsamic is applied last and in drops rather than stirred through so that its sharp, sweet intensity arrives as a distinct moment on the palate rather than disappearing into the purée. Serve immediately; the dish is complete when the purée is warm and the gamberi have not had time to carry over.