In a stand mixer, combine warm milk, sourdough discard, bread flour, softened butter, salt, honey, and one whole egg plus an egg yolk. Mix on low until incorporated, then increase to medium speed for about 5 minutes until a cohesive dough forms.

Knead the dough for approximately 15 minutes until it passes the windowpane test.

Turn the dough onto a lightly floured surface and divide it into 12 equal pieces. Shape into balls and flatten into discs.

Place the shaped discs on a lined baking sheet and let them rise for about 30 minutes to an hour until doubled in size.

Preheat oven to 350°F. Whisk together an egg white with a splash of water and brush over the buns, adding sesame seeds if desired.

Bake the buns for about 20 minutes until golden brown and they sound hollow when tapped.

Cool on a wire rack for 10-15 minutes before slicing and serving.