Place water, honey, vital wheat gluten, salt, yeast, and whole wheat flour in the bread maker in that order. Select "Dough" setting. Let the bread maker work, about 2 hours.
Remove finished dough from bread maker and punch it down on a lightly floured surface. Cut the dough into 8 equal-sized sections and pat into balls. Push a thumb through each ball to make a hole; shape the bagels as round as possible. Place on a plate and cover with a clean towel; let rest for 15 minutes.
Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with a clean towel. Sprinkle a separate baking sheet with cornmeal.
Bring a large pot of water to a boil; add sugar. Add bagels to the water, 2 at a time. Boil for 1 minute, turning them over halfway through. Remove bagels from the water with a slotted spoon and drain on the towel-lined baking sheet.
Transfer bagels to the prepared baking sheet. Brush bagels with egg white.
Bake in the preheated oven until browned, 15 to 20 minutes. Transfer to a cooling rack.