In bread machine pan, place the water, eggs, oil, honey, 1/2 cup cheese, milk powder, salt, basil, flours and yeast in order suggested by manufacturer. Select dough setting (check dough after 5 minutes of mixing; add 1 to 2 tablespoons of water or flour if needed)., When cycle is completed, turn dough onto a lightly floured surface. Shape into 12 balls. Push thumb through centers to form a 1-1/2-in. hole. Stretch and shape dough to form an even ring. Cover and let rest for 10 minutes; flatten bagels slightly., Fill a Dutch oven two-thirds full with water; bring to a boil. Drop bagels, 2 at a time, into boiling water. Cook for 45 seconds; turn and cook 45 seconds longer. Remove with a slotted spoon; drain well on paper towels., In a small bowl, combine egg white and water; brush over bagels. Sprinkle with remaining cheese. Place 2 in. apart on greased baking sheets. Bake at 400° for 15-20 minutes or until golden brown. Remove to wire racks to cool.