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1
Peel away the transparent outer membrane from each kidney. Halve lengthwise and excise the entire core of white fat and sinew — no trace of it should remain. Cut into 2 cm thick slices or retain in halves for presentation. Season on both sides. The kidney is ready when it is uniform pale rose in colour and free of any grey patches.
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2
Set a heavy sauteuse over the highest available heat until it is very hot. Add the clarified butter. Lay in the kidney pieces in a single layer without crowding. Sear for 2 minutes per side. This is where the dish lives or dies: the exterior must be mahogany brown while the interior remains distinctly pink at 55–60 °C. The moment the interior reads grey or the juices run cloudy, the kidney is irredeemable — remove it before that point.
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3
Transfer the kidneys to a warm plate to rest. Pour the Cognac into the still-hot pan and flambée; allow the flames to die completely of their own accord. The smell shifts from sharp raw alcohol to a soft, caramelised sweetness — the signal that the harsh spirit has burned off.
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4
Add the minced shallots to the pan and cook for 30 seconds, stirring, until softened. Pour in the veal jus and reduce by half over high heat. The sauce should lightly coat the back of a spoon.
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5
Remove the pan from the heat. Stir in the double cream, then the Dijon mustard. The mustard must go in off the heat; boiling it produces bitterness and destroys its pungency. The sauce should be glossy, pale ochre, and faintly pungent to the nose.
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6
Return the kidneys to the sauce and turn briefly to coat. Serve immediately in a warm cocotte or on a warm plate with the sauce pooled beneath. The surface of the kidney should still feel yielding and springy — not firm — when pressed lightly.