In a medium mixing bowl, stir together warm milk and granulated sugar until fully dissolved. Add sourdough discard and instant yeast, mixing until smooth and slightly frothy.

Gradually incorporate unbleached bread flour into the milk mixture, stirring until a shaggy dough forms. Transfer to a floured surface and knead for about 5 minutes until smooth and elastic.

Add softened butter and salt to the dough, kneading further for 5 minutes until incorporated.

Place the dough in a greased bowl, cover, and let it rise in a warm place until nearly doubled, about 1 to 2 hours.

Gently deflate the dough and divide into 8 equal pieces. Shape each piece into a smooth ball.

Let the dough balls rest for about 30 minutes, covered with a towel.

Roll each dough ball into a 7-inch log and place on a lined baking sheet.

Cover the shaped logs and let them rise until doubled in size, about 45 minutes.

Preheat your oven to 375°F (190°C).

Whisk egg whites with water to create an egg wash. Brush the top of each bun with the mixture.

Bake the buns for about 25 minutes until golden-brown.

Remove from oven and brush with melted butter. Allow to cool on a wire rack.