1. Combine Wet Ingredients: In a large mixing bowl, combine the active sourdough discard, warm milk, granulated sugar, and the first large egg. Whisk these ingredients together until they are well combined and the sugar has dissolved. The mixture will look slightly milky and might have a few small lumps from the discard, which is perfectly normal.

2. Add Dry Ingredients and Butter: Pour in the melted and cooled butter. Gradually add 3 cups of the all-purpose flour and the salt to the wet mixture. Mix with a wooden spoon or a stand mixer with a dough hook until a shaggy dough forms. It will be sticky at this point, but don’t worry.

3. Knead the Dough: Turn the dough out onto a lightly floured surface. Knead for 8-10 minutes, gradually adding the remaining 1/2 cup of flour as needed, until the dough is smooth, elastic, and no longer sticky. It should spring back 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 tightly with plastic wrap or a damp kitchen towel. Let it rise in a warm place for 4-6 hours, or until it has doubled in size. Alternatively, you can place it in the refrigerator overnight for 8-12 hours for a slower, more flavorful rise.

5. Shape the Buns: Gently punch down the risen dough to release the air. Turn it out onto a lightly floured surface and divide it into 8 equal pieces (for standard burger buns) or 12 pieces (for slider buns). Shape each piece into a tight, round ball by pulling the edges underneath and rolling it against the counter with the palm of your hand.

6. Second Rise (Proofing): Place the shaped buns on a baking sheet lined with parchment paper, leaving about 2-3 inches between each bun. Lightly cover them with plastic wrap or a clean kitchen towel. Let them rise in a warm place for 1-2 hours, or until they look visibly puffy and have increased in size by about 50%. They should feel light and airy.

7. Prepare for Baking: Preheat your oven to 375 F (190 C). In a small bowl, whisk the second large egg with a tablespoon of water to create an egg wash. Gently brush the tops of the proofed buns with the egg wash. If desired, sprinkle with sesame seeds.

8. Bake the Buns: Bake for 18-22 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.