Make the tangzhong: In a saucepan, add the flour, then while whisking, add the water and milk. Whisk constantly over medium-low heat until the mixture thickens into a smooth paste. Mixing time is approximately 3 1/2 minutes. Transfer the mixture to a stand mixer bowl and allow to cool to room temperature.
Make the dough: In a mixing bowl, add the flour and Platinum Yeast. Stir to combine.
To the lukewarm milk, add the egg, stir, then add it to the Tangzhong. Add the honey and salt; then add the flour-yeast mixture. Mix on medium-low speed with the dough hook until the mixture just comes together and then add the butter, a small piece at a time, until incorporated. Raise the mixer speed to medium.
The mixture will be very soft, and shaggy at first, sticking to the bottom and sides of the bowl. Continue mixing until the mixture pulls away from the sides of the bowl and is smooth and shiny, about 10-15 minutes. Transfer the dough to an oiled bowl and turn the dough over to coat the dough in oil.
First rise: Cover the bowl with plastic wrap and allow to rise at room temperature until the dough almost doubles in size. (See note below for overnight rise option.) Place the dough in the refrigerator to firm up for shaping, about 1/2 hour. Note: the dough is very soft and slack, but by cooling it in the refrigerator, the butter in the dough seizes and the dough firms enough to shape more easily, much like brioche dough.
Shape the rolls: Punch down dough and turn out onto a lightly greased counter. Divide dough into 16 equal pieces. Using the “claw” method, roll each bun into a tight ball. Place on a parchment-lined sheet pan, seam side down, a few inches apart (8 rolls per pan), and cover loosely with plastic wrap.
Second rise: Allow buns to rise in a warm area, no warmer than 90˚F, for 40-50 minutes or until about doubled in size and pass the ripe test (an indent remains after lightly touching dough with finger tip).
Preheat oven to 350˚F.
Bake rolls: Brush with egg wash and bake until golden brown and the internal temperature reads 190˚F, about 25-27 minutes.