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1
Crush the reserved lobster shells and heads using a heavy mallet or the back of a cleaver until broken into rough fragments. Heat the butter and oil together in a large, heavy-bottomed pot over high heat. Add the crushed shells and heads and sauté, stirring frequently, until they turn a deep orange-red throughout. The colour change confirms that the Maillard reactions have acted on the shell-surface proteins and that astaxanthin has released into the fat.
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2
Reduce the heat to medium. Add the mirepoix vegetables to the shell-flavoured fat and sweat, stirring occasionally, until softened and translucent. The pan will carry the scent of sweet onion and the underlying shell fat.
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3
Add the tomato paste directly to the pan. Cook, pressing it against the pan surface and turning it frequently, until it darkens to a brick-red and catches slightly on the base. This caramelisation of the tomato paste is where the dish's final depth of colour and sweet-savoury base either forms or fails — it must not merely warm through, and it must not scorch to black.
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4
Pour in the cognac. Tilt the pan or apply a long match to ignite the alcohol. Allow the flame to burn off completely without interference. The flame will extinguish on its own once the alcohol is consumed; the brief combustion produces the aromatic compounds that distinguish bisque from a plain shellfish soup.
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5
Pour in the white wine and raise the heat. Reduce by half, then add the warm fish stock. Bring to a gentle simmer and maintain it, uncovered, for 45–60 minutes. Over this period the residual protein in the shells fully extracts into the liquid. The bisque will deepen in colour and the surface will carry a steady, sweet-savoury shellfish aroma.
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6
Remove from the heat. Either remove the shells and blend the remaining liquid, or blend the shells together with the liquid and then pass everything through a fine-mesh strainer, pressing firmly to extract every last measure of flavour from the solid matter. The strained bisque should be smooth, with a deep orange-red colour and no visible shell fragments.
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7
Return the strained bisque to a clean pan. Add the double cream and bring to a simmer over medium heat. Reduce until the bisque reaches the desired consistency — it should coat the back of a spoon and hold a clean line when a finger is drawn across it. Adjust seasoning with fine sea salt and cayenne. Serve immediately.