Sauce Périgueux and Sauce Périgordine

Place the finely diced shallot and Madeira in a small saucepan over medium heat. Reduce by half. The liquid should be syrupy at the edges and the sharp alcohol smell should have given way to a concentrated, nutty-dried-fruit aroma.

Add the demi-glace. Simmer the two together for 5 minutes, then strain through a fine sieve into a clean pan. The sauce should be glossy, deep brown, and coat the back of a spoon.

Remove the pan from the heat and allow it to cool until the sauce is warm but not boiling. Add the diced truffle (Périgueux) or sliced truffle (Périgordine). Cover and infuse for 5 minutes. This is where the dish lives or dies: the truffle must never boil in the sauce — the volatile sulphur compounds that define its character dissolve most efficiently into warm fat and are driven off rapidly at boiling temperatures. An immediate and dramatic aromatic shift — from Madeira-and-demi-glace to something more profound and complex — should be perceptible the moment the truffle enters the warm sauce. If no such shift occurs, the truffle is of poor quality or past its peak.

Mount the sauce off heat by whisking in the cold butter cubes, a few at a time, until fully incorporated. The surface should be lightly sheen-bright, and the texture lightly thickened and smooth.

Taste the finished sauce: four notes must be present and in balance — the oxidative, nutty-dried-fruit depth of the Madeira; the savoury gelatinous richness of the demi-glace; the lactic smoothness of the cold butter; and the truffle's depth, simultaneously earthy, funky, sweet, and mineral. No single note should dominate. Serve immediately.