Bring a large pot of salted water to a boil. Add potatoes and cook until fork-tender, 12–15 minutes.
Meanwhile, melt 3 tbsp butter in a large skillet over medium heat. Add cabbage and sauté until tender, 6–8 minutes.
Stir in the garlic and cook 1 minute until fragrant. Remove skillet from heat and set cabbage aside.
Warm milk with remaining 3 tbsp butter in a small saucepan over low heat. Add green onions and let them soften, 2–3 minutes.
Drain potatoes well and return them to the hot pot. Let steam-dry 1–2 minutes for fluffier mash.
Mash potatoes. Pour in warm milk mixture gradually, mashing until creamy but not gluey.
Fold in sautéed cabbage. Season with kosher salt and black pepper to taste.
Top with parsley and an extra pat of butter if you like. Serve hot.