Add flours, sugar, and salt to bowl of a stand mixer. Stir together. Whisk egg, milk, and starter together in separate bowl until no lumps of starter remain.
Attach hook to mixer, and turn on on Kitchenaid speed 2, and gradually add liquid ingredients, until all dry bits are gone. When most of the flour is absorbed, turn mixer to speed 4 and knead for 30 seconds. Mixture will not look smooth, just shaggy. Let sit for 15 minutes.
Turn mixer onto speed 2 (using Kitchenaid Stand Mixer) and add one chunk of butter at a time to the dough, only adding another chunk once the previous is fully incorporated into the dough. Keep doing this until the butter is all added to the dough.
Once all the butter is in the dough, turn KitchenAid up to speed 4 and continue kneading for 7 minutes - the dough will be smooth, and cleaning the sides of the bowl. It will be slightly sticky to the touch still.
Overnight rise: Once kneading is done, place the dough in a bowl and cover with a damp cloth, or place bowl inside garbage bag. Leave at room temp to rise until double, about 9-10 hours if your home is at 70 degrees F. It will take longer if your home is cooler, less time if your home is warmer. *Because this is a brioche dough, it takes significantly longer to rise - don't be alarmed.
In the morning, the dough will have doubled, have a smooth top and be domed. Turn it out onto a heavily floured counter. Gently pat it into a rectangle shape and let rest for 10 minutes.
Using a floured rolling pin, roll it into a large rectangle roughly 17″x 12, making the rectangle so that one of the longest sides is closest to you.
To make the filling: Use your fingers to spread the softened butter onto the dough keeping 1/2" away from edges. This will seem odd, but this technique prevents the filling from leaking out while rising. Mix cinnamon, sugar, and salt in small bowl. Evenly distribute the mixture on top of butter.
Then use your bench scraper
to begin rolling up the dough into a log starting with the long edge closest to you. Use a little extra flour on the bench scraper each time to help pick the dough up from the surface. Don’t worry if it sticks a little, just use a little more flour dust. Roll it down to the top edge and pinch the edge to the dough log to seal it.
You should have a long log of dough roughly 17″ long. Now cut the dough log with your bench scraper, at 1 1/2 to 2" intervals (you should get 8 or 9 rolls) and place the rolls into a liberally buttered or greased 9 x 13 inch aluminum pan. (You can also use glass, but make sure it is well greased).
Place the pan inside a clean garbage bag and fold the opening over so that no air gets inside. Leave at room temp (in a warm spot if possible) to proof for 3 1/2ish hours, or about double in size. They should look like the picture above in the post, denoted with a caption that says, "after proofing, ready for oven."
When the rolls are done proofing, preheat oven to 400 degrees F. Place pan in oven, turn down to 375 degrees F, and bake for about 25-30 minutes, or until golden brown at the edges, set in the middles and smelling done.
When rolls are cooled, beat the frosting ingredients together and slather on rolls liberally.