1. Combine Wet Ingredients: In a large mixing bowl, whisk together the sourdough discard, warm milk, granulated sugar, room temperature egg, and melted and cooled butter until well combined. The mixture should look slightly milky and uniform.

2. Add Dry Ingredients: In a separate bowl, whisk together the all-purpose flour, instant yeast, 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.

3. Knead the Dough: Turn the dough out onto a lightly floured surface and knead for 8-10 minutes, or use a stand mixer with a dough hook on medium speed for 6-8 minutes. The dough should become smooth, elastic, and pass the windowpane test (you should be able to stretch a small piece thin enough to see light through it without tearing).

4. First Rise: Lightly grease a clean bowl with oil. Place the dough in the bowl, turning it once to coat. Cover the bowl with plastic wrap or a clean kitchen towel and let it rise in a warm place for 1.5 to 2 hours, or until doubled in size.

5. Shape the Buns: Gently punch down the risen dough to release the air. Divide the dough into 8 equal pieces (you can use a kitchen scale for accuracy, aiming for about 90-100 grams per bun). 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.

6. Second Rise: Loosely cover the shaped buns with plastic wrap or a clean kitchen towel and let them rise again in a warm place for 30-45 minutes, or until visibly puffy and almost doubled in size. Towards the end of this rise, 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 (1 beaten egg mixed with 1 tablespoon water). 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. Cooling: Transfer the baked buns to a wire rack to cool completely before slicing and serving. This allows the internal structure to set and prevents them from becoming gummy.