In a large bowl combine 1 1/4 c. flour, 1 1/2 c. warm water, 2T. active dry yeast, and 3T. sugar.
Mix on low speed. Scrape the sides of the bowl. Mix on high speed for 2-3 minutes.
Add 1 T. salt and 3 more cups of flour.
Mix until a stiff dough is achieved. Use the dough hook (or knead by hand on a lightly floured surface) 5-8 minutes.
Cover and let rise for 15 minutes.
Divide into 12 sections. Roll each section into a ball.
Press thumb through the center of each ball and gently pull in all directions until a uniform ring has formed.
Place on a lightly floured surface, cover, and let rest for 20 minutes.
While dough rests, grate parmesan cheese.
Bring a large pot of water to a boil. Add 1 remaining 1T. sugar and stir until dissolved.
Reduce heat to med-low.
Preheat oven to 375 degrees.
Boil 4 bagels at a time for 2 minutes on each side. Transfer each bagel to a cooling rack to drain for 15-20 seconds then transfer to a baking sheet that has been lightly sprayed with non stick cooking spray.
Repeat until all 12 bagels have been boiled.
Brush each bagel with an egg wash (1 egg + 1 T water) then sprinkle liberally with cheese.
Bake for 20 minutes. Lay a piece of foil over the pan to prevent the cheese from burning. Bake another 10 minutes. Cool on pan.