In the bowl of a stand mixer, combine the sourdough starter, warm milk, sugar, butter and eggs. Stir until well combined.
Add the salt and flour. Using the dough hook attachment, mix on medium speed for about 4 minutes.
Place the dough in a lightly greased bowl, cover it and let the dough ferment in a warm place for about 6-10 hours, or until it has doubled in size. This can vary greatly depending on the strength of your starter and the temperature of your house.
Punch down the risen dough and divide it into 8 equal pieces.
Shape each piece into a hoagie roll by rolling it into a log shape, about 6-8 inches long. It helps to first roll them out as much as you can, let them rest for a few minutes to relax the gluten and then roll them again. Otherwise the roll shapes tend to spring together.
Line a baking sheet with parchment paper and place the rolls on it.
Cover the shaped rolls with plastic wrap and let them rise at room temperature for about 4-6 hours, or until they have puffed up.
Preheat your oven to 400°F while the rolls are finishing their second rise.
Brush with egg wash.
Bake the hoagie rolls in the preheated oven for about 18-20 minutes, or until they are golden brown. Rotate them halfway.
Remove the rolls from the oven and let them cool on a cooling rack before adding your favorite sandwich fillings.