Stir together sourdough starter, butter, milk, egg, and salt. You might have to blend the butter into the batter with the back of your mixing spoon. Don’t worry if there are still some chunks of butter in the mixture.
Add the flour, 1/2 cup at a time, mixing with a spoon until it comes together enough to turn out onto a bread board or clean countertop.
Knead the dough until it is smooth and satiny, about 10 or 20 minutes.
(Optional) You can refrigerate the dough for up to 24 hours to get a more flavorful bun. Bring it back to room temperature before continuing.
Let the dough rest at room temperature 15 or 20 minutes, then roll it out to 1/2 inch thick.
Use a 3- or 4-inch biscuit cutter to cut bun shapes and transfer them to an ungreased baking sheet. You can re-roll the dough and cut more shapes until it is all used up.
(Optional) Press a light layer of sesame seeds gently into the buns.
Cover the buns lightly or put into a cool oven and let rise until doubled in bulk – between 1 and 4 hours.
Remove the buns from the oven if that’s where they were rising, then preheat the oven to 350°F. Bake the buns for 18 to 20 minutes.
Let cool on the baking sheet for 5 minutes, then transfer to a wire rack to continue cooling. Let cool completely before slicing in half horizontally.