Peel potatoes and cut into coarse chunks. Remove any tougfh outer leaves and the core from the cabbage, and finely slice the leaves.
Cook the potatoes in a large pan of simmering salted water for 10 minutes, then add the cabbage and cook until both are tender, but not overly soft, about 10 more minutes. Now add the green onions for the last minute of cooking. Drain well.
Add the butter, half and half, and most of the cheeses, leaving a little to sprinkle over the top later. Mash well, seasoning with salt and pepper.
You may serve it at this stage, sprinkled with the rest of the cheese. Or, if you like, pour it all into a buttered baking dish, sprinkle with the remaining cheese, and bake until lightly browned and steaming hot, 25-30 minutes!
If you want to make this a main dish, fry the eggs to your liking and place on top of each serving.