Make the tangzhong. Add the milk and bread flour to a small pan over low-medium heat. Whisk this mixture and as it heats, it should start to get thicker (3-5 minutes). At this point, you can use a spatula to push it around until it is a thick but light paste. Remove from the pan to a small bowl, cover, and let cool for 30 minutes.
To mix the dough, pour the milk into the bowl of a stand mixer. Add the egg, sugar, starter, and tangzhong. Use a fork to break up the mixture and mix until mostly combined. It might have clumps, but that’s fine.
Add the bread flour, all-purpose flour, and salt. Set the mixer to low-medium speed and let it combine until a moist dough forms.
Add pats of the butter one by one and let the dough consume each pat fully before adding the next pat. Then let the mixer mix the dough at medium speed for 7-9 minutes, or until the dough passes the windowpane test.
Oil a work surface and dump the dough onto the surface. Shape the dough into a ball.
Oil the mixer bowl if it’s relatively clean (or use a clean bowl) and place the dough inside. Lightly oil the top of the dough, cover with a clean kitchen towel, place in a warm place and let bulk rise for 8-10 hours.
When it’s done rising, it should be about 2.5-3 times the original size.
Lightly oil your work surface and dump the dough out onto it. (Look at that bubble!)
Press out any large bubbles and shape into a ball. Let rest for 10 minutes. Oil two 9-inch loaf/pullman pans or one 13-inch Pullman pan and set aside.
Using a lightly oiled bench scraper or sharp knife, slice the dough ball into six equal triangles, then set aside so you have some space in the center of your workspace to work.
Grab one triangle. With the triangle point facing towards you, gently flatten the dough with your fingertips into a rectangle (should flatten to resemble a strange envelope).
Pull the sides into the center.
You can stitch the sides together if they do not stay together well on their own.
Then roll up the dough from the top down (not too loose and not too tight, should just roll over itself rather naturally).
There will be tension in the dough, but don’t worry, it’s not too tight.
Slide the roll into the pan with the seam on the bottom. Repeat with each dough triangle (3 into each 9-inch pan or all 6 into the 13-inch Pullman pan), sliding the rolls in the pan to the side as you add each roll.
When you are finished shaping, cover the pan(s) with plastic or a clean kitchen towel and let them rise overnight in the fridge (10-12 hours).
Preheat the oven to 375F. When the oven has reached temperature, remove the pan(s) from the fridge. For the 9-inch pans, if you want a square loaf, leave the Pullman lids on for the entire bake. If you want a shiny, bulbous top, whisk the egg wash ingredients together and brush the loaf’s top and leave the lid off. For the 13-inch pan, you can leave the lid on and it should become bulbous without actually touching the top of the pan.
Bake for 45 minutes. If your loaves are bulbous, you can mix the sugar shiny top ingredients in a small bowl until the sugar dissolves. Set aside until the bake is done.
Remove the bread from the oven and gently overturn the loaf onto a cooling rack. Set upright. For bulbous loaves, brush the sugar shiny top mixture over the bulbous area of the loaf straight out of the oven. Let the bread cool for 2 hours before consuming for optimal crumb.