Add the first 8 ingredients to a 1-1/2- or 2-pound bread machine according to the manufacturer's directions. Select the dough cycle. When cycle is complete, remove dough from machine. Punch down. Cover and let rest for 10 minutes.
Divide dough into 24 portions. On a lightly floured surface, roll each portion into an 8-inch-long rope. Place the ropes 2 inches apart, on greased baking sheets. Cover and let rise in a warm place about 30 minutes or until nearly double.
In a small bowl combine the egg white and 1 tablespoon water; brush over breadsticks. Sprinkle with the 1 to 2 tablespoons Parmesan cheese. Bake in a 350 degree F oven for 15 to 20 minutes or until golden brown. Remove from baking sheets; cool on wire racks.