For the pretzel buns: Preheat the oven to 400 degrees F.
Form the dough into 4 buns. Make the desired slits on top of the buns and set aside.
In a large pot, combine the baking soda and 4 quarts water. Bring to a boil and then add the buns; cook for 3 minutes. Remove the buns and set on a rack to dry. Brush with the egg whites and sprinkle with pretzel salt.
Place the buns on a buttered baking sheet and bake for 12 minutes. Adjust the oven to 350 degrees F and continue baking until the buns are browned, another 15 minutes. Remove from the oven and let cool.
For the sloppy joes: In a large saute pan, heat 1 tablespoon of the olive oil over medium heat. Add the onions, peppers and garlic, and cook until slightly softened, 5 minutes. Scrape into a bowl and set aside.
Place the same skillet over high heat and add the remaining 2 tablespoons olive oil. Add the ground beef and cook until browned, breaking it up with a wooden spoon, about 6 minutes. Drain off the fat from the pan and add the chili powder, dry mustard and cayenne to the meat. Add the tomato paste, stirring to coat. Cook undisturbed for about 4 minutes to allow the tomato paste to caramelize.
Add the onion mixture back to pan and then stir in the Worcestershire, maple syrup, red wine vinegar and root beer. Reduce heat to low and simmer until the liquid has reduced and the sauce thickened, 20 minutes. Serve hot on toasted pretzel buns. (Makes enough for 9 sandwiches.)