In the bowl of a stand mixer fitted with the dough hook attachment, add the water, yeast, and sugar. Stir and let rest 5-10 minutes, until frothy.
Add the flour, salt, and melted butter, and mix using the dough hook, until combined well. Cover with plastic wrap, and allow the dough to rise in a warm place for 1 hour or until doubled in size. Punch the dough down and turn it onto a clean, lightly floured surface.
Line a large sheet pan with parchment paper or a silpat, and set aside. Cut the dough in quarters, then, cut each quarter in thirds, making 12 pieces.
Take a piece of dough and form a ball; smooth ball by pulling the sides to the center and pinching the bottom to seal.
Place the ball, pinched side down on the prepared baking sheet, with 1-inch between each roll. Cover dough balls with a clean cotton towel, and allow to rest in a warm place for 30 minutes until they rise & double in side.
Place the oven rack in the middle position, and preheat oven to 425° F.
In a large saucepan, bring 2 quarts of water to a low boil. Slowly and carefully add the baking soda, making sure that it doesn’t boil over as you add the baking soda to the water. Lower the heat to a simmer.
Place 2-3 rolls at a time into the poaching liquid, seam side down. Poach for 30 seconds. Carefully turn the roll over and poach for another 30 seconds. Remove with a slotted spoon to the lined sheet pan, seam side down. Repeat with the remaining rolls.
Using a pastry brush, brush each roll with the beaten egg, making sure to coat all sides completely. Sprinkle each roll with pretzel salt. Using a sharp straight edged knife, cut a slash or an “X” shape in the top of each roll. Bake the rolls for 15-20 minutes.
Pretzel Buns can also be made in a bread machine. Add all wet ingredients to the bowl of the machine. Add dry ingredients to top, finishing with yeast. Set machine to dough setting, and start. Once machine is done, remove dough from machine, continue with step 3.Adapted from My Life as a Mrs.