1. More than cover potatoes and carrots with water or vegetable broth in a saucepan and bring to a boil. Saute onion in oil until brown.
2. When potatoes are soft, dump out most of the cooking water except for the last couple of inches, and dump in the onions. Set aside a few chunks of potatoes and some of the onions, then blend the big pot of potatoes and onions with a hand-held blender or mixer, or use a food processor or blender, being very careful not to splash!
3. Add more water, vegetable broth, or milk as needed for desired soup-like consistency. Mash the set-aside potatoes a little, then add them and the onions back to the big pot and stir. Add salt and pepper, to taste.