While that mixture is cooling, put the water, softened butter, honey, salt, and an egg, into your mixer.
Mix these ingredients to incorporate.
Then, add the remaining flour into the mixing bowl, and mix to combine until no flour is left.
Add the sourdough starter and mix to combine.
Cover, and let this mixture sit for about 15 minutes. This is the autolyse period and it allows the fresh milled flour to absorb the liquid and the bran to soften.
After the autloyse period, then add the paste (make sure it is not hot to the touch first, or it will kill your sourdough culture and will result in an uprisen dense loaf), mix to incorporate.
Once the paste is added, Start the kneading process. Make sure not to knead the dough with the cover on your mixer, this will overheat your dough, and cause it to break down.
Knead until the dough is nice and stretchy. Typically, 6-8 minutes in a Bosch style mixer, 16-20 minutes in an Ankarsrum style mixer, or 25-30 minutes in a Kitchen aid mixer (allow extra time for a Kitchen Aid style mixer to have breaks in between kneading to prevent overheating of the mixer.) These kneading times will vary for everyone, this is just the most common times for these style of mixers.
Once the dough is nice and stretchy, and doesn't easily tear. Then, cover and allow it to rise until almost double. This can take somewhere between 3-6 hours. (In a warm area the dough will rise closer to 3 hours, in a cooler area it will take longer to rise. If the area is too cool (under about 75*F it will take a very long time to rise if it rises at all.)
While the dough is rising, prepare your bread pans with parchment paper, or softened butter.