1. Activate the Yeast: In a large mixing bowl, combine the warm milk, granulated sugar, and active dry yeast. Stir gently and let it sit for about 5 minutes until the mixture becomes foamy. This shows your yeast is active and ready to work.

2. Combine Wet Ingredients: Add the sourdough discard and the large egg to the yeast mixture. Whisk everything together until well combined. The mixture will look a bit shaggy and wet at this stage.

3. Mix Dry Ingredients and Form Dough: In a separate bowl, whisk together the all-purpose flour and salt. Gradually add the dry ingredients to the wet ingredients, mixing with a wooden spoon or a stand mixer with a dough hook until a shaggy dough forms.

4. Knead and Add Butter: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes until it’s smooth and elastic. Alternatively, if using a stand mixer, knead with the dough hook for 3-4 minutes. Gradually incorporate the melted butter into the dough during the last minute of kneading until it’s fully absorbed and the dough is soft and pliable.

5. Shape the Buns: Divide the dough into 6-8 equal pieces (depending on your desired bun size). Shape each piece into a smooth, round ball by pulling the edges underneath and pinching them together at the bottom. Place the shaped buns on a baking sheet lined with parchment paper, leaving some space between them. Gently flatten each bun slightly with the palm of your hand.

6. Quick Rest and Preheat: Loosely cover the buns with a clean kitchen towel and let them rest for 10 minutes. While they rest, preheat your oven to 375 F (190 C).

7. Egg Wash and Bake: In a small bowl, whisk together the egg yolk and water for the egg wash. Brush the tops of the buns generously with the egg wash. If desired, sprinkle with sesame seeds. Bake for 15-18 minutes, or until the buns are golden brown on top and sound hollow when tapped on the bottom.

8. Cool Before Serving: Transfer the baked buns to a wire rack to cool completely before slicing and serving. This prevents them from becoming soggy and allows the texture to set.