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

2. 2. Add Dry Ingredients: In a separate bowl, whisk together the all-purpose flour, salt, and instant yeast. Gradually add the dry ingredients to the wet ingredients, mixing with a wooden spoon or your hands until a shaggy dough forms.

3. 3. Knead the Dough: Turn the dough out onto a lightly floured surface. Knead for 5-7 minutes until the dough is smooth, elastic, and no longer sticky. It should spring back when gently poked. If it’s too sticky, add a tiny bit more flour, a tablespoon at a time.

4. 4. First Rise: Lightly grease the mixing 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 to 1.5 hours, or until doubled in size.

5. 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 an oval shape, about 5-6 inches long, resembling a hot dog bun. Place the shaped buns about 1 inch apart on a baking sheet lined with parchment paper.

6. 6. Second Rise: Cover the shaped buns loosely with plastic wrap or a clean kitchen towel and let them rise again for another 30-45 minutes, or until visibly puffy.

7. 7. Prepare for Baking: Preheat your oven to 375 F (190 C). If using an egg wash, whisk together the egg yolk and water in a small bowl. Gently brush the tops of the risen buns with the egg wash for a golden, shiny crust.

8. 8. Bake the Buns: Bake for 12-15 minutes, or until the buns are golden brown on top and sound hollow when tapped on the bottom. Remove from the oven and let cool on a wire rack before slicing and serving.