In a large saucepan of boiling salted water, cook the turnip and potato cubes for 15 to 17 minutes, until they are tender, and drain them.
In a bowl, mash them with a fork and stir in the scallions, the egg, flour, and salt and pepper to taste.
Coat the bottom of a large, heavy bottomed skillet with about 1/4-inch of the oil. Heat the pan on medium high heat until the surface of the oil begins to shimmer, but not smoke.
Spoon 1/4-cup mounds of the turnip potato batter into the pan, flattening them into 1/2-inch thick patties with the back of a spatula. Fry the patties until they are golden, turning them once, about 4 minutes on each side. Transfer the patties to paper towels to drain off excess oil.