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

2. Add Melted Butter and Dry Ingredients: Stir in the melted butter. Then, add the all-purpose flour, salt, and instant yeast (if using). Mix 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 5-7 minutes, or until the dough is smooth and elastic. If using a stand mixer, knead for 3-5 minutes on medium speed. The dough should be slightly tacky but not sticky.

4. First Rise (Bulk Fermentation): Place the 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 until doubled in size. This will take about 1-2 hours with yeast, or 3-4 hours without yeast. You’ll know it’s ready when it springs back slowly when gently poked.

5. Shape the Buns: Gently punch down the risen dough to release the air. Divide the dough into 6-8 equal pieces (depending on your desired bun size). Roll each piece into a smooth, tight ball. Place the shaped buns on a baking sheet lined with parchment paper, leaving about 2 inches between each bun.

6. Second Rise (Proofing): Cover the baking sheet loosely with plastic wrap or a clean kitchen towel and let the buns rise again in a warm place for another 30-45 minutes, or until they look puffy and have nearly doubled in size.

7. Prepare for Baking: Preheat your oven to 375F (190C). 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 and sprinkle with sesame seeds, if desired.

8. Bake the Buns: Bake for 15-18 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.