In a microwave safe measuring cup, heat the milk to 100°F / 38°C. Whisk in the yeast and the sugar. Set aside until the mixture begins to foam. (~5 minutes)
While waiting, add the tangzhong, flour, salt, egg, and butter to the bowl of a stand mixer fitted with the dough hook attachment. When the yeast mixture begins to foam, add it in to the rest of the ingredients.
Knead the dough on low speed for 1 minute to allow the ingredients to come together. Increase the speed to medium-high and knead for an additional 10 minutes, or until the dough passes the windowpane test. (This will happen after the dough has cleaned the sides and bottom of the bowl)
Take the dough out of the bowl and shape it into a large ball. Lightly butter the inside of the mixing bowl, then place the dough into the bowl. Cover the top of the bowl with a towel, then leave the dough to rise in a warm environment until it has doubled in size (~1 hr.)
Once the dough has doubled in size, lightly punch it down to remove any excess air.
Butter a medium sized rectangular pan (9x7") for 8 rolls. If you only have an 8x8" pan, that will work perfectly to make 9 rolls.
Weigh the dough then divide it into 8 or 9 equally sized rolls, depending on your pan size listed above. Roll each piece of dough tightly into a ball, then place the rolls evenly spaced apart in the buttered pan.
Cover the top of the pan tightly with plastic wrap, then place it in the fridge to rest overnight.
Take the rolls out to sit on the counter 30 minutes before you would like to bake them.
Preheat the oven to 350°F / 175°C. Make an egg wash by whisking together 1 egg and 1 tbsp. of milk, then brush it over each roll.
Bake the rolls for 15-20 minutes, until the tops are golden brown and the internal temperature reads 190°F / 88°C.