In the bowl of a stand mixer, add the active dry yeast and sugar to the warm milk. Stir gently and let it sit for 5 to 7 minutes. You're looking for a foamy, bubbly top and a yeasty smell. If it doesn't foam, your milk was too hot or your yeast is old. Start over before going any further.
Add the sourdough discard (room temp or cold from fridge), 1 egg, and softened butter (not melted) to the yeast mixture. Mix until roughly combined. The butter doesn't need to be perfectly incorporated yet.
Add the salt and the all-purpose flour. Mix on low until a shaggy dough forms, then increase speed and knead for 8 to 10 minutes. If the dough is too sticky or wet, add one tablespoon of flour at a time until a dough forms. You want a soft, slightly tacky dough, not a stiff one.
Place the dough in a lightly greased bowl and cover with plastic wrap or a damp towel. Let it rise in a warm spot until doubled in size, about 1 to 90 minutes depending on how warm your kitchen is.
Punch down the dough and divide it into 8 equal pieces. Shape each piece into a smooth ball by pulling the dough underneath itself and pinching it closed at the bottom. Place on a parchment-lined baking sheet, spacing them a few inches apart. Gently flatten each ball slightly with your palm.
Cover the shaped buns loosely and let them rise again for 30-60 minutes, until puffy and nearly doubled.
Preheat your oven to 375°F. Brush each bun with the egg wash made from 1 egg (for egg wash, whisked with a splash of water). Sprinkle with your toppings if you are using any.
Bake for 18 to 20 minutes, until deep golden brown on top. Let them cool on the pan for at least 10 minutes before slicing. They firm up as they cool.