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

2. Add Dry Ingredients: To the wet mixture, add the salt, all-purpose flour, and instant yeast. Mix with a wooden spoon or a stand mixer with a dough hook on low speed until a shaggy dough forms.

3. Knead the Dough: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes until it becomes smooth, elastic, and no longer sticky. If using a stand mixer, continue kneading with the dough hook for 5-7 minutes on medium-low speed. The dough should spring back when gently poked.

4. First Rise: Place the kneaded dough in a lightly oiled 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 60-90 minutes, 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. Roll each piece into a smooth, tight ball. Place the shaped buns on a baking sheet lined with parchment paper, leaving about 2-3 inches between each bun.

6. Second Rise: Cover the baking sheet loosely with plastic wrap or a clean kitchen towel and let the buns rise again in a warm place for 30-45 minutes, or until visibly puffy and almost doubled in size.

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 with sesame seeds.

8. Bake the Buns: Bake for 15-20 minutes, or until the buns are golden brown on top and sound hollow when tapped on the bottom. Remove from the oven and transfer to a wire rack to cool completely before slicing and serving.