On a lightly floured surface, roll the dough into a circle about 1/8 inch thick. Cut out as many circles as possible with a 3½-to 4-inch cookie cutter. Then gather the remaining scraps into a ball, roll out again, and cut out additional circles. With a pastry brush, coat each circle with a light film of the beaten egg white. Drop 1 tablespoon of the cheese or 1 teaspoon of fruit on the lower half of each circle. Bring the exposed half of the circle up over the filling and press all around the dough firmly with the back of a fork. Make certain that the edges are thoroughly sealed to prevent the filling from seeping through. Set aside, loosely covered with a towel, until ready to cook.
Bring 4 quarts of salted water to a boil in a 5- to 6-quart pot and drop in 6 dumplings and gently stir a few times. Lower the heat and simmer the dumplings uncovered for 8 to 10 minutes, or until they float to the surface of the water. Remove with a slotted spoon and transfer to a heated platter. Cover loosely with foil to keep them hot while you cook the remaining dumplings similarly.
To serve, arrange the dumplings on a heated platter and moisten them with the hot melted butter. If you have used the fruit filling, serve the dumplings with cherry or blueberry juice. The cheese-filled dumplings are customarily served with the remaining cup of sour cream.