Make Dough: Add all the ingredients to the bowl of your mixer, starting with ¾ cup of water, and mix until the dough is elastic and shiny, about 5 minutes. Add more water as needed, until the dough is smooth, soft and comes clean from the sides of the bowl.
Rise: Transfer the dough to a greased bowl, cover with plastic wrap or a damp clean towel and allow it to rise in a warm, draft free environment until doubled in size.
Prep Pan: Lightly grease your New England hot dog pan.
Shape Dough: Cut the dough into 10 equal pieces. To be precise, you can weigh the dough, this way you'll get 10 equal portions. Roll out each piece so that it's about 4x6-inches, then tightly roll up the shorter side to form a tube shape. Pinch the seam to seal it and then place it sealed side down, in the prepared pan. Repeat with remaining pieces.
Rest: Cover the pan with plastic wrap and place them in a warm, draft free environment. Let them rest for another 45 min to an hour or until doubled in size.
Preheat Oven: Preheat the oven to 350°F degrees.
Bake: Brush the buns with the egg wash and transfer the pan to the preheated oven. Bake for 15-20 minutes or until golden brown.
Cool: Let the buns cool enough so you can handle them comfortably. Remove them from the pan and let them finish cooling on a wire rack. Gently pull the buns apart before storing them or using them.