Combine the starter, water, milk, butter, honey, one egg, and salt in a mixer until well incorporated.
Add the flour one cup at a time until the dough beings to clean the sides of the bowl and form a ball (start watching for this at around 3 cups of flour and add more from there). Knead for 10 minutes until the dough is soft and smooth.
Let the dough rise for 6-8 hours or until close to double. You can also place in the fridge for 1-3 days for a long fermentation.
Shaping the hot dog buns:Divide the dough into 12 pieces (about 3 ounces each). Roll into balls. Let the dough rest for about 30 minutes covered. This allows the dough to relax so it doesn't bounce back when shaping. There are two options for shaping hot dog buns. Try both and see which you prefer!Technique 1:Stretch the dough and roll it into a bun shape. It should be 5-6" long. Longer and skinnier is better than short and squat to get that hot dog bun shape.Technique 2: Stretch the dough into a flat 4x6" disc. Roll it into a log, pinch the seams closed. Let the buns rise, covered, on a baking sheet. Keep them somewhat close and the buns will be pull-apart when done baking, just like store bought.
Preheat the oven to 350 degrees. In a small bowl, beat one egg, gently brush the tops of the buns with the egg wash.
Bake for 16-18 minutes until golden. Cool and enjoy!