Melt half of the butter over moderate heat. Add onion and cook, stirring occasionally, until onion is golden and starting to brown at the edges. (8 to 10 minutes)
Add the chicken broth, potatoes, and water. Bring to a boil, reduce the heat to a simmer, and cook, partially covered, until potatoes are soft. (20 to 25 minutes)
Mash potatoes just enough to thicken soup, but leave plenty of chunks intact. Add the milk and remaining butter.
Add shredded cheese. If soup needs to thin out, add additional chicken broth.
Season the soup with salt, pepper, garlic, nutmeg, and cayenne. Test and add additional spices if needed.
Chop fresh chives and add to the soup.
Serve hot in bowls with additional cheese and green onions and Salt SGO.