Combine all of the dough ingredients (except the cheese) in the bowl of a stand mixer and beat on medium speed for 10 minutes or until the dough becomes shiny and satiny. It will be very sticky. Scrape down the sides of the bowl periodically during mixing.
Add the cheese to the dough and continue to mix until it is totally combined.
Scrape the dough into a lightly greased bowl and cover with plastic wrap. Allow the dough to rest in a warm, draft-free place (approximately 1 hour). No need to worry that it didn't rise much. It won't at this point.
Turn the dough out onto a lightly floured surface and roll the dough out into a log shape. Divide the dough into 6 equal portions. Set three portions aside while you shape the first loaf.
Roll three portions of dough into approximate 9" ropes. Braid the three ropes together, pinching the ends. Place the braid into a greased loaf pan, making sure to tick the ends under if needed. Continue the same process with the second loaf.
Allow the braided loaves to rise until considerably larger (this will take approximately 3½ - 4 hours).
Preheat oven to 425 degrees.
Brush the top of the risen loaves with the reserved egg whites and sprinkle with additional shredded cheese.
Bake the bread in a preheated oven for 15 minutes then reduce the heat of the oven to 350 degrees and continue baking for 30-35 minutes until the loaves are a deep, golden brown.
Remove and allow to cool completely before slicing.