Make the dough: Start by warming the milk, water and butter together in the microwave (about 1 minute).
Add the sourdough starter and sugar to a mixing bowl with the paddle attachment and mix to combine. Slowly add the warmed milk mixture then slowly add the bread flour and salt, about 2 minutes. It should be a wet dough.
Scrape down the sides, cover with the towel and let it rest for 30 mins. (don't forget to replenish your starter!)
Switch to the dough hook and knead for about 6 minutes until the dough becomes smooth. You could also knead it with your hands on a gently floured surface.
Bulk rise: Transfer the dough to a greased bowl and cover with a towel. Place in a warm spot to rise for 8 hours.
Shape and 2nd rise: Gently grease a 9x13 inch baking pan. Remove the dough from a lightly floured surface and gently shape the dough into a rectangle. Divide the dough into 12 pieces (or 8 if you want bigger buns). Gather the ends and fold into the middle, then pinch the dough together.
Flip the dough over and gently roll to give the buns a soft surface. Transfer them all to the prepared baking pan.
Cover and rise for another 30 mins - 1 hour (dough will expand but not double)
Bake: Whisk together the egg and water then brush the tops with egg wash then bake at 400°F for 25-30 minutes, or until the buns are a deep golden brown. Brush gently with butter or vegan butter if desired. Serve warm or at room temperature with your burgers. They will last up to 2 days!