1. 1. Activate the Dough Base: In a large mixing bowl, combine your active sourdough starter, warm milk, granulated sugar, and one large egg. Whisk these ingredients together until well combined. The mixture should look slightly milky and have some small bubbles from the starter.

2. 2. Add the Fats and Dry Ingredients: Pour in the slightly cooled melted butter. Then, add 3 1/2 cups of all-purpose flour and the salt. If you’re using the optional instant or active dry yeast, add it now. Mix with a wooden spoon or a dough whisk until a shaggy dough forms and most of the flour is incorporated.

3. 3. Knead the Dough: Turn the shaggy dough out onto a lightly floured surface. Knead for 8-10 minutes until the dough is smooth, elastic, and no longer sticky. It should spring back when gently poked. If the dough is too sticky, add a very small amount of flour, a tablespoon at a time, until it reaches the right consistency.

4. 4. First Rise (Bulk Fermentation): Lightly grease a clean bowl with oil. Place the dough in the bowl, turning it once to coat the top. Cover the bowl tightly with plastic wrap or a damp kitchen towel. Let it rise in a warm place for 2-4 hours, or until doubled in size. Alternatively, you can refrigerate it overnight for a slower, more flavorful rise.

5. 5. Shape the Buns: Gently punch down the risen dough to release the air. Divide the dough into 8-12 equal pieces, depending on your desired bun size. For each piece, gently pull the edges of the dough towards the center and pinch to seal, then flip it over and cup your hand around the dough, rolling it on the counter to form a smooth, tight ball. This creates tension on the surface, which helps the buns rise nicely.

6. 6. Second Rise (Proofing): Line a baking sheet with parchment paper. Arrange the shaped dough balls on the prepared baking sheet, leaving some space between them as they will expand. Cover loosely with plastic wrap or a light kitchen towel. Let them rise in a warm place for 1-2 hours, or until they look puffy and have nearly doubled in size. They should feel light and airy.

7. 7. Prepare for Baking: Preheat your oven to 375 F (190 C). In a small bowl, whisk the remaining egg with a tablespoon of water to create an egg wash. Gently brush the tops of the risen buns with the egg wash. If desired, sprinkle sesame seeds over the tops for a classic burger bun look.

8. 8. Bake to Golden Perfection: Bake the buns for 15-20 minutes, or until they are golden brown on top and sound hollow when tapped on the bottom. Once baked, transfer the buns to a wire rack to cool completely before slicing and serving. Cooling is crucial for the texture to set properly.