In a large bowl, whisk warm milk, sugar, and instant yeast. Let sit 5 minutes until frothy. Whisk in melted butter, egg, and sourdough discard.

Add flour and salt. Stir to a shaggy dough, turn onto a floured surface, and knead 8 to 10 minutes until smooth. Place in a greased bowl, cover, and let rise in a warm place until doubled, about 1 hour.

Punch down dough. Divide into 8 equal pieces and shape into tight balls. Place on a parchment-lined baking sheet, spaced 2 inches apart.

Cover loosely and let rise until puffy and nearly doubled, 45 to 60 minutes.

Preheat oven to 375°F (190°C). Beat the egg for wash with 1 tbsp water and brush over buns. Sprinkle with sesame seeds.

Bake 18 to 22 minutes, rotating halfway, until deep golden brown. Cool on a wire rack at least 15 minutes, then slice and serve.