Firstly, make sure the night before you feed your sourdough starter heavy, so it is nice and strong for this recipe. (Even if you already fed it in the morning. You want it nice and strong the night before.)
Then, mill the flour. I like to use mostly hard white wheat with a little Kamut (Khorasan Wheat). Just make sure to use mostly hard wheat of some sort for this recipe. Otherwise, it will not develop the gluten enough to have a nicely risen loaf of bread.
In a stand mixer, add water, softened butter, sugar (or honey), salt, and egg. Mix all these ingredients to combine.
Then, add the freshly milled flour to the stand mixer, and start to mix slowly until all the dry flour is completely incorporated.
Add the sourdough starter into the dough, and mix until incorporated. (The dough will seem very wet at this stage.)
Cover, and let the dough sit at room temperature for a minimum of 15 minutes, or up to 1 hour. This resting time is important, It allows the fresh milled to start absorbing the liquid, and the bran to start softening. This is the Autolyse step.
After the dough has had time to Autolyse, then start the kneading process. This kneading process could take anywhere between 7 to 30 minutes. This time varies greatly, so the best way to tell if the dough is done being kneaded is to check the dough itself. You want a nice stretchy dough that wants to stay together. This is a wet dough, so try to resist adding flour to it.
Once kneaded, cover the dough and let it rise until almost doubled. This can take anywhere from 2 to 8 hours depending on your sourdough starter, and how warm it is in your kitchen. (Other variables I mentioned in my Sourdough Sandwich Bread blog post HERE can affect this rise time as well.) Mine generally takes about 3- 4 hours for example.
Prepare your baking sheet with parchment paper towards the end of the first rise.
After the dough has almost doubled in size, then on a lightly oiled surface, divide the dough into 12 equal pieces.
For Hamburger Buns: Roll each dough ball into a ball, making sure to gently pull some surface tension onto the tops of the dough ball. (the dough should feel more tight with the surface tension. I have a short video to help.) Then, lightly flatten the buns into a disk shape. Place on parchment paper lined baking sheet, and repeat with the rest of the hamburger/sandwich buns.
For Hot Dog Style Buns: Flatten each piece and roll it up longwise, then pinch the ends. Roll each dough log until it is about 5-6 inches long. Making sure there is some surface tension. (I have a short video showing this method.) Place these sourdough hot dog buns seam side down on a parchment paper lined baking sheet. Then, repeat with the rest of the dough pieces.
Lightly spray the buns with some oil, and cover with plastic wrap.
Let these fresh milled flour sourdough buns rise this second time for 2-3 hours, or until it has risen and looks puffy. (Times may vary greatly, so make sure to check your dough occasionally during this second rise time.)
During the end of this second rise preheat oven to 350*F.
Make the egg wash by mixing 1 egg, and a tsp of water. Get your toppings ready if using any.
Once the buns have risen and look puffy, you can gently brush with the egg wash to make the buns shiny, more browned, and allow the toppings to stick. This step is optional.
Bake buns for 20-25 minutes until the internal temperature of the center of the buns are 205-210*F. If they are below this temp, they may be gummy in the center.
Take out of the oven when the the center reaches the desired temp range. Then, remove them from the baking pans to a wire rack to cool.
Let the buns cool completely before cutting them! If you cut into them before they cool completely, they may be gummy on the inside.
Once cooled completely, then slice, and enjoy!