Coarsely grate the potatoes and onions. Place in a colander and press down to drain off as much liquid as possible. Transfer to a large bowl and mix in the salt, pepper and baking powder. Add the egg yolks and mix well.
In a medium bowl, beat the egg whites until they hold their shape. Fold the egg whites into the potato mixture. Fold in the flour.
In a large skillet, add enough oil to measure half an inch. Heat over medium flame until it begins to shimmer. Working in batches, carefully drop large tablespoonfuls of the potato mixture into the hot oil; press down lightly with the back of a spoon to flatten; do not crowd the pan. Cook until golden, about 3 minutes per side.
Transfer to paper towels to drain. Serve piping hot with applesauce and sour cream on the side.