Cover potatoes with cold water in a saucepan; add 1/4 teaspoon salt. Bring to a boil; reduce heat. Simmer until tender, 10 to 15 minutes; drain. Return to pot; cover.
Stir together horseradish, vinegar, parsley, oil, and a pinch of salt; set aside.
Bring stock, wine, celery, and garlic to a boil in a tall, narrow 6-quart pot. Reduce heat to medium-low; add leeks and 3 sprigs thyme. Simmer until vegetables are tender, 5 to 10 minutes. With a slotted spoon, transfer vegetables to a large bowl; cover. Place beef and peppercorns in pot; add water just to cover, if necessary. Partially cover pot; cook at a bare simmer until an instant-read thermometer reads 130 degrees.when inserted into center of beef, 20 to 25 minutes. Remove beef, and let rest 10 minutes.
Carve beef on a carving board into 18 thin slices; add any juices to broth. Pour broth through a fine sieve into bowl with vegetables; discard solids. Divide soup, potatoes, and beef among six bowls; garnish with thyme. Serve sauce on the side.