Activate the yeast — In the bowl of your stand mixer, combine the yeast, sugar, and warm milk. Give it a gentle stir and leave it to proof for about 5 minutes — you're looking for a foamy, fragrant mixture that tells you the yeast is alive and ready to work.
Add dry ingredients — Add the bread flour and salt to the yeast mixture. Using the dough hook attachment, mix on low speed until the dry ingredients are just incorporated.
Mix in the egg and fat — Add one egg and the melted butter (or avocado oil) to the bowl. Continue mixing with the dough hook until the dough comes together into a rough, shaggy mass. Don't worry if it looks a bit scraggly at this point — that's exactly right.
Rest, then knead — Let the dough rest for 5 minutes, then knead with the dough hook for another 5 minutes until smooth and elastic.
First rise — Cover the bowl tightly with plastic wrap or a snug lid and leave the dough to rise for 30 minutes.
Shape the buns — Punch down the dough to release any trapped air, then transfer to a lightly floured surface. Divide into 8 equal pieces. Roll each into a log about 6 inches long, pinch the sides and ends to seal, and place seam-side down on a parchment-lined baking sheet.
Second rise — Cover loosely and let the buns rise for 15 minutes until puffy.
Egg wash and bake — Preheat your oven to 350°F. Whisk the remaining egg and brush each bun generously. Bake for 12 to 15 minutes until deep golden brown. Let cool a few minutes before slicing.