In a large mixing bowl (or stand mixer bowl), combine the warm milk, granulated sugar, and active dry yeast. Stir gently and let sit for 5-10 minutes until the yeast is foamy. This indicates it's active.

Add the sourdough discard, melted butter, and egg to the yeast mixture. Whisk until well combined.

Gradually add 3 cups of the all-purpose flour and the salt. Mix with a wooden spoon or the dough hook on low speed until a shaggy dough forms. Add the remaining 1/2 cup flour, a little at a time, until the dough comes together and is not overly sticky. You may not need all of it or might need a tiny bit more.

If using a stand mixer, increase speed to medium-low and knead for 8-10 minutes until the dough is smooth, elastic, and pulls away from the sides of the bowl. If kneading by hand, turn the dough out onto a lightly floured surface and knead for 10-15 minutes.

Lightly grease a clean bowl with oil. Place the dough in the bowl, turning once to coat. Cover tightly with plastic wrap or a clean kitchen towel. Let rise in a warm place for 2-3 hours, or until doubled in size. The exact time will depend on the temperature of your room.

Gently punch down the dough to release the air. Turn it out onto a lightly floured surface. Divide the dough into 8 equal pieces (approx. 95-100g each for large buns, use a kitchen scale for accuracy).

Shape each piece into a tight, smooth ball. To do this, flatten a piece slightly, then pull the edges into the center and pinch to seal. Flip it over and cup your hand around the dough, rolling it on the counter to create tension and a smooth, round shape.

Line a large baking sheet with parchment paper. Arrange the shaped buns on the baking sheet, leaving about 2-3 inches between them.

Cover the buns loosely with plastic wrap or a clean kitchen towel. Let them rise again in a warm place for 45-60 minutes, or until visibly puffy and almost doubled in size.

Preheat your oven to 375°F (190°C) during the last 15 minutes of the second rise.

In a small bowl, whisk together the egg and water for the egg wash. Gently brush the tops of the proofed buns with the egg wash. If desired, sprinkle with sesame seeds.

Bake for 15-20 minutes, or until the buns are golden brown on top and sound hollow when tapped on the bottom. If they start to brown too quickly, you can loosely tent them with foil.

Remove from the oven and transfer the buns to a wire rack to cool completely before slicing and serving.