1. Activate the Yeast and Combine Wet Ingredients: In a large mixing bowl, combine the warm milk, granulated sugar, and active dry yeast. Stir gently and let it sit for 5-10 minutes until the mixture becomes foamy. This indicates the yeast is active. Then, add the sourdough discard, lightly beaten egg, and melted butter to the foamy mixture. Whisk everything together until well combined.

2. Mix in Dry Ingredients to Form a Dough: Gradually add the all-purpose flour and salt to the wet ingredients. Start by adding about half the flour and mix with a wooden spoon or spatula until a shaggy dough forms. Continue adding the remaining flour, mixing until the dough comes together and pulls away from the sides of the bowl. It might still be a bit sticky, which is perfectly normal.

3. Knead the Dough Until Smooth and Elastic: Turn the dough out onto a lightly floured surface. Knead the dough for 8-10 minutes. At first, it will be sticky, but as you knead, it will become smoother and more elastic. You’ll know it’s ready when it passes the “windowpane test” – you can stretch a small piece of dough thin enough to see light through it without tearing.

4. First Rise (Bulk Fermentation): Lightly grease a clean large bowl with olive oil. Place the kneaded dough into the bowl, turning it once to coat all sides with oil. Cover the bowl tightly with plastic wrap or a clean kitchen towel. Let the dough rise in a warm place for 1.5 to 2 hours, or until it has doubled in size. The dough should look visibly puffy and airy.

5. Shape the Buns and Prepare for Second Rise: Gently punch down the risen dough to release the air. Turn it out onto a lightly floured surface and divide it into 8 equal pieces. Roll each piece into a tight, smooth ball. Place the shaped buns on a baking sheet lined with parchment paper, leaving about 2-3 inches between each bun. Gently flatten each bun slightly with the palm of your hand.

6. Second Rise (Proofing): Cover the baking sheet loosely with plastic wrap or a clean kitchen towel. Let the buns rise in a warm place for another 30-45 minutes, or until they look noticeably puffy and have increased in size by about 50%. While they are rising, preheat your oven to 375 F (190 C).

7. Egg Wash and Bake: Just before baking, gently brush the tops of the risen buns with the egg wash (egg yolk mixed with water). This will give them a beautiful golden-brown color and a slight sheen. If desired, sprinkle sesame seeds over the tops. Bake for 15-18 minutes, or until the buns are golden brown and sound hollow when tapped on the bottom.

8. Cool Before Serving: Once baked, transfer the buns to a wire rack to cool completely. Allowing them to cool ensures they set properly and prevents them from becoming gummy inside. Serve warm or at room temperature.