1. 1. Activate the Yeast: In a large mixing bowl, combine the warm water, active dry yeast, and granulated sugar. Stir gently and let it sit for 5-10 minutes until the mixture becomes foamy. This foaminess indicates that your yeast is active and ready to work its magic.

2. 2. Combine Wet Ingredients: To the activated yeast mixture, add the unfed sourdough discard, melted butter, and one large egg. Whisk everything together until it’s well combined and smooth.

3. 3. Mix in Dry Ingredients: Gradually add the salt and 3 1/2 cups of all-purpose flour to the wet ingredients. Mix with a wooden spoon or a stand mixer with a dough hook until a shaggy dough forms. If the dough is still very sticky, add the remaining 1/2 cup of flour, a little at a time, until it pulls away from the sides of the bowl.

4. 4. Knead the Dough: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes. The dough should become smooth and elastic. Alternatively, if using a stand mixer, knead on medium speed for 5 minutes.

5. 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 60-90 minutes, or until it has doubled in size.

6. 6. Shape the Buns: Gently punch down the risen dough to release the air. Divide the dough into 8-12 equal pieces, depending on whether you want burger-sized buns or smaller slider buns. Shape each piece into a smooth, tight ball by pulling the edges underneath and rolling it against the counter with the palm of your hand.

7. 7. Second Rise and Preheat Oven: Arrange the shaped buns on a parchment-lined baking sheet, leaving some space between them. Cover loosely with plastic wrap or a kitchen towel and let them rise again for 30-45 minutes, or until visibly puffy. During this time, preheat your oven to 375 F (190 C).

8. 8. Egg Wash and Bake: Just before baking, gently brush the tops of the buns with the beaten egg wash. If desired, sprinkle with sesame seeds or poppy seeds. Bake for 15-20 minutes, or until the buns are golden brown on top and sound hollow when tapped on the bottom.

9. 9. Cooling: Transfer the baked buns to a wire rack to cool completely before slicing and serving. This allows the steam to escape and prevents them from becoming soggy.