Peel and rinse the potatoes. Grate potatoes and the onion with a wire grater (this gives a nicer texture than the food processors). Add the flour, egg, salt, and pepper. You want the mixture to be thick enough to hold together when you drop a spoonfull into the water. Drop about a tablespoon full into salted boiling water. When the dumplings start to float cook about 5 more minutes. Serve with melted butter.
[NOTE: Finding a wire grater, if you don't already have one, can be something of a challenge. Try an old time hardware store. If you find them, be sure to buy more than one because they may not be available anymore when the first one breaks a wire or two.]