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1
Kill and split each lobster immediately before cooking. Remove the claws from the body. Place the split halves flesh side up, shell side down, on a grilling tray. Grill the halves under a maximum-heat element for 6–8 minutes. At the correct endpoint, the flesh is opaque throughout but retains a slightly creamy, translucent appearance at the deepest point of the tail; the shell is deep orange-red. Press the flesh — it yields and springs back. This is 75% cooked. The flesh will pass through two further cooking stages; it must not be taken further here.
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2
While the lobster halves grill, poach the claws in rapidly boiling water for 8 minutes. Drain, crack the shells, and extract all claw meat cleanly.
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3
Carefully extract the tail flesh from the grilled half-shells. Cut into medallions. Reserve the cleaned shells.
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4
Soften the shallots in butter over a medium heat without colouring. Deglaze with the white wine, then add the Cognac and flambé. Reduce until almost dry. Add the cream and reduce to a coating consistency. Remove the pan from the heat entirely before stirring in the Dijon mustard, fresh tarragon leaves, and a squeeze of lemon juice. A correctly made sauce will simultaneously carry the aromas of the sea, cream, tarragon, and the sharpness of mustard — all four notes perceptible at once.
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5
Add the lobster medallions and claw meat to the sauce — just enough to coat and warm through. Spoon the mixture back into the cleaned half-shells. Mask the surface with additional sauce.
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6
Scatter the finely grated Gruyère evenly over the filled shells. Place under a maximum-heat grill with the surface no more than 10 cm from the element for 4–5 minutes. The dish is ready when the surface is gold and bubbling, the Gruyère showing its own Maillard spotting, and the sauce at the edges still moist and cream-coloured beneath the crust.