In a large bowl, combine the wet ingredients: sourdough starter, egg, butter, and milk.
Add the dry ingredients: instant yeast, salt, sugar, and bread flour. Mix until a shaggy dough forms.
Knead for 8 minutes with a stand mixer on low speed (dough hook attached) or 10 minutes by hand, until the dough only slightly sticks to the bowl’s bottom. A stand mixer can make handling sticky, enriched dough easier. Resist the urge to add extra flour; your dough should be soft and stick slightly to the bowl’s bottom. A wetter dough creates lighter, softer buns, while too much flour results in tougher buns.
Cover the bowl with a damp towel and let the dough rise for about 1 hour and a half, or until doubled.
On a clean surface, divide the dough into 8 large, 10 regular-sized, or 12 smaller portions. Use a bench scraper or sharp knife for clean cuts.
Flatten each piece into a rectangle, roll it up tightly, and seal the edges. You can use your hands or a rolling pin. If the dough starts sticking to the counter too much, try greasing the surface with a small amount of oil instead of adding more flour. Extra flour can make the buns tougher.
Place the shaped buns in a baking pan, cover, and let rise for about 1 hour, until puffy and light.
Before baking, brush each bun with an egg wash. Add your favorite toppings if desired; sesame seeds, poppy seeds, or everything bagel seasoning work great here!
Bake at 400°F for 15-20 minutes, or until golden brown. Transfer buns to a cooling rack for 15 minutes before slicing. You can cut them along the side or across the top, depending on your preference.