Combine porcini mushrooms and water in a bowl; soak until soft, 1 hour. Drain and reserve liquid. Dice and set mushrooms and reserved liquid aside.
Preheat oven to 325 degrees F.
Season beef generously with salt and pepper. Heat vegetable oil in a large ovenproof skillet over high heat. Place beef in the skillet; cook until brown on all sides, 5 to 8 minutes per side.
Reduce heat to medium and stir in 1 tablespoon butter, shallots, and a pinch of salt; cook until shallots are soft and translucent, 5 to 7 minutes.
Pour tarragon vinegar into the skillet and bring to a boil while scraping any browned bits off of the bottom. Stir until liquid is reduced by half, 2 to 4 minutes.
Pour in veal stock, cream, 1/2 cup reserved mushroom liquid, mushrooms, salt, and pepper; stir to combine. Return beef to the skillet.
Roast in the preheated oven until meat is medium rare, about 45 minutes. An instant-read thermometer inserted into the center should read 130 degrees F (54 degrees C). Transfer meat to a plate and loosely tent with foil. Set skillet over high heat and bring pan juices to a boil.
Stir in 1 tablespoon butter and tarragon; season with salt and pepper to taste. Return tenderloin and any accumulated juices to skillet and serve.