In a 4- or 5- quart Dutch oven melt butter over medium heat. Add onion, salt, and pepper. Cook, stirring occasionally, until onion is translucent but not brown. Add potatoes, 2 cups of the water, and rosemary. Bring to boiling; reduce heat. Simmer, covered, for 20 minutes or until potatoes are tender when pierced with a fork. Remove from heat.
With a potato masher, mash potatoes until no lumps remain. Or for a smoother soup, transfer soup, a portion at a time, to a blender or food processor. Cover and blend or process until nearly smooth. Return all to Dutch oven. Add remaining 2 cups of water. Bring to boiling; reduce heat. Simmer, uncovered, about 10 minutes more or until thickened and desired consistency, stirring frequently to prevent soup from sticking to bottom of pan.
While soup is simmering, in a 10-inch skillet heat oil over medium-high heat. Add carrot ribbons a few at a time to hot oil. Cook for 1 to 2 minutes until crisp. Transfer to a cooling rack or paper towels; immediately sprinkle with kosher salt.
Ladle soup into bowls, then top with crisp carrots and a sprinkle of coarse pepper.