Scrub potatoes with vegetable brush under cold running water. Peel. Cut into eighths. Cut onions into chunks. Place in food processor container fitted with metal blade. Process until no large pieces remain. Turn into strainer and drain very well.
Pour potato/onion mixture into large bowl. Add carrot, egg, salt and pepper. Stir in just enough flour to make mixture hold together.
In large skillet over medium-high heat, heat 1 inch of oil. Carefully drop about 1/4 cup potato mixture into hot oil, flattening slightly. Fry until golden brown, about 3 to 5 minutes on each side. Repeat.