1. Combine Wet Ingredients and Discard: In a large mixing bowl, whisk together the warm milk, sourdough discard, granulated sugar, one large egg, and melted butter. Ensure the milk isn’t too hot, or it could kill the yeast.

2. Add Dry Ingredients (Except Flour): Stir in the salt and instant yeast until well combined. The mixture should look slightly frothy if your yeast is active.

3. Gradually Incorporate Flour: Add 3 cups of the all-purpose flour to the wet mixture. Mix with a wooden spoon or a stand mixer with a dough hook until a shaggy dough forms. Gradually add the remaining 1/2 to 1 cup of flour, a little at a time, until the dough comes together and is no longer sticky, but still soft. You might not need all the flour.

4. Knead the Dough: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes, or use a stand mixer with a dough hook for 3-5 minutes, until the dough is smooth and elastic. It should spring back when gently poked.

5. 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 1 to 1.5 hours, or until doubled in size.

6. Shape the Buns: Gently punch down the risen dough to release the air. Divide the dough into 8 equal pieces (for standard burger buns). 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.

7. Second Rise and Preheat Oven: Cover the shaped buns loosely with plastic wrap or a kitchen towel and let them rise again for 30-45 minutes, or until puffy and almost doubled in size. During the last 15 minutes of the second rise, preheat your oven to 375 F (190 C).

8. Egg Wash and Bake: 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. Bake for 15-18 minutes, or until the buns are golden brown on top and sound hollow when tapped on the bottom. Let them cool on a wire rack before slicing and serving.