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1
Dispatch each lobster with a swift, firm knife thrust through the cross behind the head. Section the lobsters raw: separate the claws, cut the tail through the shell into 5–6 medallions, and split the head in half lengthwise. Reserve all tomalley and coral in a small bowl. Season the pieces.
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2
Heat the olive oil in a wide, heavy sauté pan to 210 °C — the surface will shimmer and the first wisp of smoke will appear. Add the lobster pieces in a single layer and sauté for 2–3 minutes, turning once. The dish lives or dies here: the shells must turn a vivid, uniform scarlet and the exposed flesh must show a firm, white edge before the liquid enters the pan; underdone at this stage, the sauce will lack its essential depth.
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3
Pour the Cognac over the lobster pieces. Tilt the pan toward the flame or touch a lit match to the vapour at the pan's edge and stand clear. Allow the flames to subside completely — the brandy is spent when the fire dies of its own accord and the sharp alcohol smell gives way to a sweet, caramelised fragrance.
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4
Add the minced shallots and garlic to the pan and cook for 1 minute, stirring, until softened and translucent but not coloured.
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5
Add the dry white wine, fish fumet, tomato concassée, bouquet garni, and cayenne. Bring to a gentle simmer, cover, and braise for 15–18 minutes. The lobster medallions are done when they yield with slight resistance to light finger pressure through the shell.
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6
Lift the lobster pieces from the pan. Extract the tail meat and crack the claws to remove the meat intact. Set aside, covered, in a warm place.
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7
Discard the bouquet garni. Reduce the braising liquid over moderate heat until it has halved in volume — it will coat a spoon lightly and deepen in colour.
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8
Remove the pan from the heat entirely. Mash the reserved tomalley and coral with the 30 g of softened butter to a smooth paste and whisk it into the hot sauce in one motion. Follow immediately with the 20 g of cold butter, piece by piece, swirling to emulsify. The sauce must not return to a boil — if it does, the liver proteins will turn grainy and bitter. The finished sauce is glossy and just thick enough to coat the back of a spoon.
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9
Scatter the fresh tarragon leaves into the sauce. Return the lobster meat to the pan and turn gently to coat. Serve immediately in deep, warmed plates alongside rice pilaf.