Combine wet ingredients: In a large bowl, whisk together sourdough discard, warm milk, one egg, and sugar until smooth.
Add dry ingredients: Stir in bread flour, whole wheat flour, instant yeast, and salt until a shaggy dough forms. Let rest 10 minutes to hydrate the whole wheat flour.
Incorporate fat: Mix softened tallow (or butter) into the dough until fully incorporated.
Knead the dough: Knead by hand for 8–10 minutes or in a stand mixer for 5–6 minutes, until the dough is soft, slightly tacky, and stretchy.
Bulk ferment: Place dough in a greased bowl, cover, and let rise until doubled in size (about 60–90 minutes).
Divide and shape: Turn dough onto a lightly floured surface. Divide into 8–10 pieces (depending on desired bun size). Shape each portion into a tight ball by tucking the edges underneath.
Final proof: Place buns on a parchment-lined baking sheet, flatten slightly, cover, and let rise until puffy and about 1½ times their original size.
Bake: Preheat oven to 190°C / 375°F. Brush buns with egg wash and sprinkle with sesame seeds. Bake 15–18 minutes until golden brown.
Cool: Let buns cool at least 30 minutes before slicing to set the crumb.