1. Activate Your Starter and Combine Wet Ingredients: In a large mixing bowl, combine your active, bubbly sourdough starter with the warm water, granulated sugar, and salt. Stir gently until the sugar and salt are mostly dissolved and the mixture is well combined. The mixture should look slightly milky and have a few small bubbles from the active starter.

2. Incorporate Butter and Flour: Add the melted unsalted butter to the wet ingredients. Gradually add 3 cups of the all-purpose flour, mixing with a wooden spoon or a stand mixer with a dough hook until a shaggy dough forms. Continue adding flour, a little at a time (up to 4 cups total), until the dough comes together and pulls away from the sides of the bowl. It should be soft but not overly sticky.

3. Knead the Dough for Development: Turn the dough out onto a lightly floured surface. Knead for 8-10 minutes until the dough is smooth, elastic, and passes the windowpane test (you should be able to stretch a small piece thin enough to see light through it without it tearing). This develops the gluten, which is crucial for fluffy buns.

4. First Rise (Bulk Fermentation): Place the kneaded dough in a lightly oiled bowl, turning it once to coat. Cover the bowl with plastic wrap or a damp kitchen towel and let it rise in a warm spot for 4-6 hours, or until it has roughly doubled in size. You’ll see visible air bubbles and the dough will feel soft and pillowy.

5. Shape the Hot Dog Buns: Gently punch down the risen dough to release the air. Divide the dough into 8-10 equal pieces (about 80-90 grams each for standard hot dog buns). Roll each piece into a smooth ball, then flatten it slightly into an oval. Roll the oval tightly into a log shape, pinching the seam closed at the bottom. Taper the ends slightly.

6. Second Rise (Proofing): Place the shaped buns seam-side down on a baking sheet lined with parchment paper, leaving about 2 inches between each bun. Cover loosely with plastic wrap or a clean kitchen towel and let them rise again for 1-2 hours, or until they look noticeably puffy and light. They should spring back slowly when gently poked.

7. Prepare for Baking: Preheat your oven to 375 F (190 C). If using, gently brush the tops of the proofed buns with the beaten egg wash. Sprinkle with sesame seeds if desired.

8. Bake to Golden Perfection: Bake for 15-20 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.