1Whisk and cook flour, milk, and water in a saucepan over medium heat until smooth and thick like pudding.
2Transfer the Tangzhong to a bowl, press plastic wrap against the surface, and let cool.
3Mix warm milk with a teaspoon of sugar and a packet of yeast, and let it sit for about 5-10 minutes until frothy.
4Add the Tangzhong, egg, sugar, flour, and salt to the frothy yeast mixture.
5Mix the ingredients into a shaggy dough.
6Knead the dough with a dough hook for about 5 minutes.
7Add in the melted butter and knead for another 5 minutes until the dough is soft, smooth, and no longer sticking to the sides.
8Lightly coat the dough with flour, shape it into a nice ball, cover with plastic wrap, and let it proof for 1 to 2 hours until it has doubled in size.
9After the first proof, divide the dough into 9 pieces.
10Shape each piece into a dough ball.
11Place the dough balls in a greased or parchment-lined square pan.
12Cover the pan tightly with plastic wrap and let the dough balls proof a second time until doubled (about another hour).
13Preheat the oven to 3500F (1750C) and brush the rolls with an egg wash.
14Bake the rolls for about 20 minutes, tenting with foil if they are browning too fast.
15Once out of the oven, cool them in the pan for a few minutes, then lather them in some melted butter.