Mix together 10 grams of ripe and active sourdough starter with 100 grams flour and 100 grams water. Cover and let rise overnight or about 12 hours until it has many bubbles, doubled in size and peaked.
To the bowl of a stand mixer add levain, milk, egg, sugar, most of the bread flour and salt. Mix until completely combined and add the rest of the flour a little bit at a time until the dough is tacky (not super sticky) and pulls away from the sides of the bowl. Knead together for 3-5 minutes until smooth. Note: If you already have ripe, bubbly, active sourdough starter on hand you can substitute 200 grams of it for the levain in this recipe.
Cut the softened butter into 4 pieces and add them one at a time into the center of the dough as it mixes. Continue kneading for another 5 minutes until all the butter is incorporated and the dough is smooth and shiny.
Put the dough in a container and set it in a warm place for 4-5 hours. Dough temperature should be 78ºF during this bulk fermentation. If your dough is colder than 78ºF, you will want to extend bulk fermentation. If the dough is warmer, bulk fermentation will be a little shorter.
At the end of bulk fermentation, the dough will have puffed up a little (maybe 30%) and feel smooth and airy. If it hasn't done these things, let it sit for a little longer. Note: At this point, you have the option to place the dough in the refrigerator overnight or for up to 48 hours. If you decide to refrigerate the dough, you may need to add a little extra time for the buns to rise once shaped.
Dump the dough out onto a countertop. Cut the dough into 8 equal pieces, about 130 grams each. Take each piece of dough and pull/pinch up the sides until it forms a ball. Roll the ball on the counter using your hand in a cupping shape (see video here) to seal the balls and create tension for the roll to rise.
Line a large baking sheet with parchment paper and place dough balls on the baking sheet apart from each other. Cover the dough and let the buns rise until doubled in size, about 3-4 hours if kept at 78-80ºF. I like to set my covered dough on a proofing mat or in my oven with the light turned on. This keeps the buns in a warm (78-80ºF) environment to rise.
You will know the burger buns have risen when they are puffed up and feel light and airy. The dough will not be hard or stiff. When you push in on the dough it will leave a small indentation in the dough, springing back just a little bit. If the dough springs back completely and the dough feels hard or compact, let it rise another half hour and check on it again. You may need to warm up the dough if it doesn't seem to be rising much.
Once the buns have risen, preheat the oven to 400ºF (remove the buns from the oven first if they are in there for rising). Place a baking stone on the bottom rack of the oven to prevent the bottom of the buns from burning.
Mix together the egg with a splash of water and brush egg wash on top of the buns if desired. Sprinkle with your favorite topping, ie: sesame seeds or "everything but the bagel" seasoning.
Bake for about 18-20 minutes until light brown, baked through and the inside of the bun registers 190º-195ºF. Let cool completely before toasting or using as burger buns. Enjoy!
