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Combine the milk and bread flour in a medium saucepan. Place over medium heat and cook, stirring continuously, until the mixture thickens into a paste. Remove from heat and transfer to the bowl of a stand mixer.
Add the cold milk and mix to combine, checking with your finger that it is not hotter than lukewarm (the cold milk should cool the hot tangzhong enough). Add the sugar, yeast, milk powder, salt, egg, and flour. Transfer to the mixer and fit with the dough hook.
Mix the dough on medium speed until it is smooth and elastic and clearing the sides of the bowl, about 12-15 minutes. Don’t freak out, as it is sticky - if you have made it by weight you will be fine. Set a timer and walk away from the mixer if you need. If after that time it really isn’t coming together and you’re worried, add flour a teaspoon at a time just until the dough just comes together.
Add the butter and mix for a further 5 minutes until incorporated. The dough should be smooth and elastic, and pass the windowpane test.
Turn the dough out onto a surface and flour very lightly if needed to bring into a tight ball with a bench scraper. Transfer to a greased bowl and cover the bowl with plastic wrap.
Place the dough in a warm spot and rise until doubled in size, approximately 1 ½ hours.
Turn the dough out onto a lightly floured surface. If you are using rings for your buns (I used 100mm rings for 90g buns), grease them lightly.
Divide the dough into 9 equal portions, each weighing about 90g. Working with one piece of dough at a time, flatten out the piece of dough, then tuck up into a ball, then turn the ball seam side down and roll into a tight ball by cupping your hand to create a 'claw' shape, using the tension from the counter to roll the dough tightly. Place to the side and cover lightly with plastic wrap. Repeat with the remaining pieces of dough, grouping the balls together on your counter with a little space between them so they don't touch.
Leave the buns to rest for 10 minutes, then give them a quick roll to tighten them back up. Place them in the prepared rings or spaced apart on the baking sheets. They rise quite a lot so ensure they are far enough apart - you can do 5 on on pan and 4 on the other. Alternatively you can space them closer so they bake to be touching.
Cover the buns either with a lid, or some lightly greased plastic wrap, or place a second sheet pan upside down over the top to act as a lid. Leave the buns to rise again for about 1 ½ hours. You want them to puff up and double in size, and when you press lightly on one, it should leave a small indentation that doesn’t quite spring back. See images for how they look just after rolling and then risen. Remember that rising time depends on your environment so go by how the dough is looking, rather than a rising time.
When there is about 20 minutes to go in the rise, preheat the oven to 360°f / 185°c. Brush the buns with egg wash. If you are baking in rings just brush what is exposed, if you are baking plain buns brush the edges too. and bake in the oven for 15-18 minutes, until they are a deep golden brown. Remove from the oven and remove the rings if using.
Leave to cool on the pans for 10-15 minutes then transfer to a wire rack and allow to cool completely. The buns will wrinkle slightly as they cool and if you used the rings the edge may not stay completely straight and may fold on itself a little - this is due to them being very soft and is totally normal.