Peel potatoes, rinsing and setting in a bowl or bowls of water as you go (potatoes can stay in water up to 2 hours). When you’re ready to cook, drain potatoes, cut into 2- to 3-in. chunks, and put in a 5- to 6-qt. pot with water to cover.
Bring pot to a boil over high heat, then reduce heat and simmer until potatoes are tender when pierced, about 15 minutes. Drain.
In bowl of a stand mixer, beat part of potatoes at a time until smooth; or mash by hand in a large bowl. As mashed, transfer to a very large bowl and cover with foil.
Put cabbage and 1/2 cup butter in pot used for potatoes (no need to wash). Cover and cook over medium-high heat, stirring often, until cabbage is well wilted, 5 to 8 minutes. Meanwhile, in a medium saucepan, heat milk and green onions until simmering, stirring often, 5 to 8 minutes.
Stir cabbage mixture and three-quarters of milk mixture into potatoes until blended. Then stir in more milk mixture if you’d like a looser texture. Season to taste with salt and pepper. Spoon into a serving bowl and top with a couple of big pats of butter and more pepper if you like.
Make ahead: Let cool completely, then chill airtight up to 1 day. Microwave in batches, loosely covered, until hot, stirring occasionally.