Combine yeast and sugar in a small bowl. Pour warm water over the top and let proof for 5-10 minutes or until the yeast looks foamy and bubbling.
In a stand mixer with your dough hook, add melted butter, 1 cup water, 3 cups of flour, and salt. When your yeast is foamy, pour it into the bowl. Mix until combined, 1-2 minutes.
The dough should be soft, but not sticky. It should pull away from the edges of the bowl when mixing. If it is too sticky, add in flour 1/4 cup at a time. When the dough is soft, let the mixer knead the dough for 5-6 minutes to help create gluten which gives you a softer, chewier end result.
Place dough in a lightly-greased bowl. Cover with plastic wrap and let rise for about one hour.
Punch dough down and place on a clean, floured surface. Divide into eight pieces. This will make eight large full-size hoagie rolls. You can divide into 16 pieces for half-size rolls.
Shape each piece of dough into a rectangle by placing your hands on each side of the dough and moving your hands together in a vertical motion, and then in a horizontal motion to get the dough tucked underneath itself. When you have the shape you want, place the roll onto a baking sheet lined with parchment paper. You can use a bread lame to make slashes across the top if you want.
Cover the rolls with clean towel, and let them rise for another 30 minutes. In a small bowl, whisk the egg and 1 tablespoon cold water together. Using a pastry brush, carefully brush the egg wash on the top of each roll. This will give you a nice golden brown shiny top.
Preheat oven to 375℉. Bake for 15-20 minutes for small rolls, and 20-25 for larger rolls or until the top of the loaves are golden brown.
Remove from oven to a cooling rack and allow them to cool completely before cutting them. I recommend using a quality bread knife to slice them.
Add your favorite sandwich toppings and enjoy!