Peel potatoes; rinse. Submerge in water until all are peeled. Drain; cut into 2-inch pieces. Place potatoes in pan; cover with water. Bring to a boil, covered, over high heat. Reduce heat; simmer until tender when pierced, about 15 minutes. Drain. Mash, part at a time in a heavy-duty mixer or by hand in a large bowl. As mashed, transfer to a very large bowl. Place cabbage and butter in pan used for potatoes. Cover, steam cabbage over medium-high heat until well wilted, stirring often. Meanwhile, stir milk and onions often in a 3-4 quart pan over medium heat until simmering, 8-10 minutes. Stir milk mixture into potatoes until smooth; stir in cabbage and Cheddar cheese. Season with salt and pepper.