1. In a large bowl, mix the starter and milk until combined, then whisk in eggs. Set aside. 
2. In a separate bowl, whisk together the flours, sugar, and salt. Add this to the milk mixture and stir until combined. Mix in the softened butter. 
3. Transfer the dough to a stand mixer fitted with a dough hook, and set to low speed. Mix the dough for about 5 minutes, until the texture begins to transform from craggy to smooth and the dough starts to pull away from the sides of the bowl. It will still be very sticky. 
4. Lightly oil a medium bowl and scrape the dough into it. Cover with plastic or a damp towel and allow to rest at room temperature. 
5. After 15 minutes, perform one set of stretch-and-folds, lifting one side of the dough toward you and folding it back in over itself, then turning the bowl a quarter-turn and repeating this process three more times. Allow the dough to rest 15 minutes more, than perform another set. Do this one last time, for a total of 3 sets of stretch-and-folds over 45 minutes.
6. Bulk proof:  Allow the dough to rest an additional 3.5 – 4 hours, until its volume has increased by roughly 1/3 and looks smooth and firm.
7. Butter a 9×13″ or 10×10″ pan. Tip the dough out onto a floured counter and, using a sharp knife or bench scraper, cut the dough into 15 (if using 9×13″ rectangular pan) or 16 (if using a 10×10″ square) even pieces. To ensure even size, you can use a kitchen scale to weigh the pieces. 
8. Form the rolls:  for each piece of dough, fold the sides into the middle, creating a rough ball, and pinch it to seal. Flip the roll so it’s seam-side down and, keeping the seam on the bottom, gently spin it with your palms, tucking it under with the edges of your pinky fingers, to create a little surface tension. 
9. Place the rolls evenly in the prepared pan, leaving a little space in between each. Cover with plastic. At this point you can 1) allow the rolls to rise at room temperature for 2-3 hours until they have doubled and nearly reached the top of the pan, or 2) give them an extended/overnight rise in the fridge (recommended for optimal flavor development). *Note:  If you choose the fridge option, be sure to allow around 3 extra hours the next day to let the rolls fully rise at room temperature once you remove them from the fridge.
10. If you’re using the egg wash (which is not necessary but will give the rolls a nice, shiny finish), whisk your egg in a small bowl with a little bit of water. When the rolls are fully risen, lightly brush the tops. Preheat the oven to 375 degrees F. 
11. Bake at 375 degrees F for 35-40 minutes, until the rolls are deep golden brown. Let them cool slightly, then brush the tops with melted butter (again, not necessary but oh so delicious). 

Wrap any leftovers tightly in plastic to avoid drying out.