1. Activate Your Starter and Combine Wet Ingredients: In a large mixing bowl, combine your active sourdough starter, warm water, sugar, salt, and olive oil (or melted butter). Stir these ingredients together until well combined and the sugar and salt have dissolved. The mixture should look slightly milky and uniform.

2. Gradually Add Flour to Form Dough: Begin adding the all-purpose flour, one cup at a time, mixing with a wooden spoon or a stand mixer with a dough hook. Continue adding flour until a shaggy dough forms. It should be sticky but start to pull away from the sides of the bowl. You might not need all 4 cups of flour; stop when the dough is manageable.

3. Knead the Dough Until Smooth: Turn the dough out onto a lightly floured surface. Knead for 8-10 minutes until the dough is smooth, elastic, and no longer sticky. It should spring back slowly when gently poked. If using a stand mixer, knead on medium speed for 6-8 minutes.

4. First Rise (Bulk Fermentation): Lightly grease a clean bowl with a little oil. Place the dough in the bowl, turning it once to coat. Cover the bowl with plastic wrap or a damp kitchen towel and let it rise in a warm place for 2-4 hours, or until doubled in size. The exact time will depend on the strength of your starter and the ambient temperature.

5. Shape the Hot Dog Buns: Gently punch down the risen dough to release the air. Turn it out onto a lightly floured surface and divide it into 8-10 equal pieces, depending on your desired bun size. Roll each piece into an oval shape, about 5-6 inches long, resembling a hot dog bun. Place the shaped buns on a baking sheet lined with parchment paper, leaving about 1-2 inches between each bun.

6. Second Rise (Proofing): Cover the shaped buns loosely with plastic wrap or a damp towel and let them rise again in a warm place for another 45-60 minutes, or until visibly puffy and almost doubled in size. While they are proofing, preheat your oven to 375 F (190 C).

7. Prepare and Apply Egg Wash (Optional) and Bake: If using, whisk together the beaten egg and milk in a small bowl. Gently brush the tops of the risen buns with the egg wash. This will give them a beautiful golden color. Bake for 15-18 minutes, or until the buns are golden brown and sound hollow when tapped on the bottom.

8. Cool Before Serving: Transfer the baked buns to a wire rack to cool completely before slicing and serving. This allows the internal structure to set and prevents them from becoming gummy.