In a medium mixing bow stir together 3/4 cup of the flour, the 1/3 cup Parmesan cheese, the yeast, and Italian seasoning. Add the 3/4 cup warm water and the oil to dry mixture. Beat with an electric mixer on low to medium speed for 30 seconds, scraping side of bowl constantly. Beat on high speed for 3 minutes. Using a wooden spoon, sir in as much of the remaining flour as you can.
Turn dough out onto a lightly floured surface. Knead in enough of the remaining flour to make a stiff dough that is smooth and elastic (8 to 10 minutes total). Shape into a ball. Place dough in a lightly greased bowl, turning once to grease the surface. Cover; let rise in a warm place until double in size (45 to 60 minutes).
Punch dough down. Turn out onto a lightly floured surface. Divide into 4 portions. Divide each portion into 6 pieces. Cover; let rest for 10 minutes. Grease baking sheets. To shape breadsticks, roll each piece into an 8-inch-long rope. Place ropes 2 inches apart on prepared baking sheets. Cover; let rise in a warm place until nearly double in size (about 30 minutes).
Stir together egg white and the 1 tablespoon water; brush on breadsticks. Sprinkle with additional Parmesan cheese. Bake in a 375 F oven for 10 minutes. Reduce oven temperature to 300°F Bake for 20 to 25 minutes more or until golden and crisp. Transfer to wire racks; cool completely.