Combine 2 cups flour, onion, 2 tablespoons sugar, yeast and salt in large mixing bowl. Add 1-1/2 cups warm water. Beat on low speed of electric mixer 1 minute or until blended. Increase speed to high; beat 3 minutes.

On lightly floured surface, knead in enough remaining flour to make moderately stiff dough. Cover with plastic wrap; let rest 10 minutes.

Spray 2 baking sheets with nonstick cooking spray; set aside. Divide dough into 24 equal portions. Shape each portion into ball. Press thumbs through center of each ball. Stretch to form 1-1/2-inch hole. Place on prepared baking sheets. Cover with plastic wrap and clean cloth. Let rise in warm place 30 minutes or until doubled in bulk.

Preheat broiler. Broil bagels about 5 inches from heat source 2 to 3 minutes, turning once. (Bagels should not brown.) Bring 6 cups water and remaining 1 teaspoon sugar to a boil in Dutch oven. Drop 7 or 8 bagels into boiling water mixture. Gently simmer 3 minutes, turning once. Drain on paper towels. Repeat with remaining bagels.

Preheat oven to 375°F. Place bagels on baking sheets sprayed with cooking spray. Beat egg white and remaining 1 tablespoon water in small bowl. Brush tops of bagels with egg white mixture. Sprinkle with Parmesan cheese. Bake about 20 minutes or until golden brown.