Cover potatoes with water in a small pot; season with salt. Bring to a boil over medium-high heat, then reduce heat and simmer until a paring knife slides easily through the flesh, 30–40 minutes. Drain, let cool slightly, and peel.
Meanwhile, melt 4 Tbsp. butter in a large saucepan over medium heat. Add leeks and cook, stirring frequently, until very soft, 8–10 minutes. Add garlic and cook, stirring frequently, until garlic is fragrant and leeks are just beginning to brown around the edges, about 3 minutes longer. Add 1 cup cabbage and cook, stirring constantly, until wilted. Add milk and cream and bring to a simmer.
Add potatoes and remaining 1 cup cabbage, then coarsely mash with a potato masher. Season with salt and pepper.
Transfer colcannon to a large serving bowl. Top with remaining 2 Tbsp. butter