Begin by melting 1 tablespoon of rendered goose fat in a reduction pan over low heat. Ensure the heat is gentle to avoid burning.

Add the finely chopped shallots to the melted goose fat. Cook them over low heat until they become softened. Be cautious not to let the shallots brown; the goal is to achieve a soft texture.

Pour in 2 teaspoons of truffle liquid from the preserved jar. Add the truffle peelings, 2 tablespoons of madeira, 1/2 cup beef stock, 1 teaspoon tomato paste, and 1 tablespoon veal glaze. Stir the mixture and let it simmer over low heat for approximately 15 minutes, allowing the flavors to meld.

Remove the pan from heat and strain the sauce through a fine mesh strainer. Use a wooden spoon to gently press against the sauce, helping it pass through the strainer effectively.

Wipe out the reduction pan and return the strained sauce to it.

Season the sauce with 1/4 teaspoon of salt and a pinch of freshly ground black pepper. Introduce the thinly sliced black truffle to enhance the dish's earthy aroma.

Bring the sauce to a light boil and immediately reduce the heat to a low simmer. Allow the mixture to reduce by about a third, which should take approximately twenty minutes.

Remove the pan from heat and stir in 1 tablespoon of sweet, unsalted butter and 1 tablespoon of finely chopped fresh parsley. This adds richness and a burst of fresh flavor to the sauce.

Transfer the prepared Perigourdine sauce to a sauceboat. Serve it while still warm, complementing your favorite dishes. Enjoy the luxurious and truffle-infused flavors!