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1
Dispatch each lobster with a swift, firm knife thrust through the cross on the head. Immediately separate the claws, cut the tails into 4–5 medallions through the shell, and split the heads. Extract and reserve all coral and tomalley raw. The lobster pieces are ready when every cut is clean and the coral is intact.
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2
Heat the olive oil and butter together in a wide, heavy sauté pan over high heat until the butter foams and begins to subside. Add the lobster pieces shell-side down in a single, uncrowded layer. Sear without disturbance until the shells turn bright red and the exposed flesh at the cut edges caramelises to a deep golden colour — the Maillard reaction on the shell proteins is the foundation of the sauce's depth. Turn once to colour the flesh side briefly.
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3
Pour the cognac over the lobster pieces and ignite immediately. Tilt the pan to sustain the flame and baste the pieces continuously until the flame dies of its own accord and the alcohol is consumed. The pan is ready when silence returns and the cognac aroma has mellowed to a rounded, integrated note.
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4
Add the shallots and garlic to the pan and stir for one minute. Add the tomate concassée, tomato paste, white wine, fish stock, cayenne, and tarragon. Stir to combine, scraping up any caramelised residues. Cover and simmer over moderate heat for 15–18 minutes. The lobster medallions are done when the flesh is just set and opaque throughout — 18 minutes is the ceiling; beyond it the meat toughens irreversibly.
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5
Lift all lobster pieces from the pan and set aside, loosely covered. Raise the heat and reduce the sauce, uncovered, until it has fallen by half and coats the back of a spoon with body.
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6
Mash the reserved raw coral and tomalley thoroughly into the 50 g softened butter until a smooth, homogeneous beurre de corail is formed. Remove the reduced sauce from the heat. Whisk the beurre de corail into the hot — not boiling — sauce in small additions, incorporating each piece fully before adding the next. This is the step on which the dish lives or dies: the coral must remain raw until this moment so that it cooks and emulsifies as it enters the sauce; if the sauce boils after the coral is added, the emulsion breaks to a grainy, split mass. The finished liaison is a glossy, coral-orange emulsion of perceptible body.
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7
Strain the sauce through a fine-mesh sieve, pressing lightly. Return the lobster pieces to the pan and warm gently in the strained sauce without allowing it to simmer. Arrange the pieces on a warmed platter, nap with the sauce, and finish with fresh tarragon leaves. The dish is ready when the sauce clings to each piece with a glossy sheen and the aroma of tarragon, cognac, and shellfish is vivid and unified.