In a large mixing bowl stir together 1/2 cup all-purpose flour and the yeast. Add water, honey, oil, and salt. Beat with an electric mixer on low speed for 30 seconds, scraping sides of bowl constantly. Beat on high speed for 3 minutes. Using a wooden spoon, stir in whole wheat flour, 1 tablespoon Parmesan cheese, and 1/2 teaspoon oregano.
Turn the dough out on a lightly floured surface. Knead in enough of the remaining all-purpose flour to make a moderately soft dough that is smooth and elastic (6 to 8 minutes total). Shape into a ball. Place in a lightly greased bowl, turning once to grease surface. Cover and let rise in a warm place until double (about 1 hour).
Punch dough down. Turn dough out onto a lightly floured surface. Cover and let rest for 10 minutes. Lightly grease two baking sheets.
Roll dough into a 15x7-inch rectangle. Cut into fifteen 1-inch strips. Place strips on prepared baking sheets. (If desired, twist each strip 3 to 4 turns, stretching gently to form 10-inch-long twisted breadsticks. Place on prepared baking sheets and press ends to baking sheet.) Cover and let rise in warm place until nearly double (about 30 minutes). Brush breadsticks with egg white. In a small mixing bowl stir together 1 tablespoon Parmesan cheese and 1/2 teaspoon oregano. Sprinkle over breadsticks.
Bake in a 375 degree F oven about 12 minutes or until golden brown. Remove from baking sheets. Cool on wire racks. Makes 15 breadsticks.