Place the potatoes and balsamic vinegar into a large pot and cover with 2 to 3 inches of water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer for about 12 minutes. Place cabbage in a metal colander, then place the colander into the pot with the potatoes. Continue simmering until the potatoes are tender, about 8 minutes more. Remove the colander of cabbage, drain, and set aside. Drain potatoes and allow to steam dry for a minute or two. Place potatoes in a large bowl. Mash the potatoes with butter and milk until smooth. Set aside.