Heat 1 tablespoon of the oil in a large skillet over medium heat. Add the shallots, thyme, and ¼ teaspoon each salt and pepper. Cook, stirring occasionally, until the shallots are tender, 4 to 6 minutes. Remove from heat and remove the thyme sprigs. Add the vinegar and butter and stir until melted. Transfer to a small bowl.
Wipe out the skillet and heat 1 teaspoon of the remaining oil over medium heat. Season the steak with ½ teaspoon each salt and pepper and cook, 3 to 5 minutes per side for medium-rare. Let rest for 5 minutes before serving.
Meanwhile, heat the remaining tablespoon of oil 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 mixture and top with the shallots.