Heat 1 tablespoon of goose or duck fat in a large skillet over medium heat. Thinly slice the shallots and place them with the thyme, and ¼ teaspoon each salt and pepper into the pan. Cook, stirring occasionally, until the shallots are tender, 4 to 6 minutes. Remove from the heat and remove the thyme sprigs. Add the vinegar and butter to the pan and stir until melted. Transfer everything to a small bowl.
Wipe out the skillet and heat 1 teaspoon of goose or duck fat over medium heat. Season the steaks with ½ teaspoon each of salt and freshly ground black pepper and cook, 3 to 5 minutes per side for medium-rare. Remove from the heat and allow the steaks to rest for 5 minutes in a warm place before serving.
While the steaks are cooking, wash the cremini mushrooms, trim the stems and thinly slice. Heat 1 tablespoon of goose or duck fat in a second large skillet over medium-high heat. Add the mushrooms and ¼ teaspoon each salt and pepper and cook, tossing occasionally, until browned, 4 to 6 minutes. Add the watercress and cook, tossing, until tender, 2 to 4 minutes more. Serve the steak with the watercress and mushroom mixture and top with the shallots.