How to Make It
Preheat the oven to 425°. Heat the goose fat in a large, nonstick skillet over moderate heat until hot but not smoking. Add the mushrooms, season lightly with sea salt, and sauté just until the mushrooms begin to give up their juices, about 5 minutes. Reduce the heat, add the thyme, garlic, half the parsley and half the walnuts. Cook, stirring regularly, for about 1 minute more. Season generously with sea salt and pepper.
Transfer the mixture to a 9-inch tarte Tatin pan or oven proof, nonstick skillet. Place the Tatin pan or skillet on a baking sheet. Remove the pastry from the refrigerator and place it on top of the mushroom mixture, gently pushing the edges of the pastry down around the edge of the pan. Bake the tart on the baking sheet in the center of the oven until the pastry is golden, 20 to 25 minutes.
Meanwhile, in a small jar, season the lemon juice with sea salt. Cover and shake to dissolve the salt. Add the walnut oil, cover, and shake again. Taste for seasoning.
Remove the baking sheet from the oven. Immediately invert a serving platter with a lip over the skillet. Quickly but carefully invert the tart on top of the serving platter so the mushrooms are on top and the pastry is on the bottom. Should any mushrooms stick to the bottom of the pan, remove them and place them back into the tart. Sprinkle the mushrooms with the remaining tablespoon of parsley and the remaining walnuts. Drizzle with 2 teaspoons of the walnut oil vinaigrette (reserve the remaining vinaigrette for another use); season with fleur de sel. Serve warm, cut into wedges.