In a small heavy pan, whisk flour, milk, and water together until smooth. Bring to a simmer over medium-low heat and cook, whisking often, until thickened but still pourable, about 10 minutes. The tangzhong will thicken more as it cools. Scrape into a cup or bowl and lightly cover the surface with plastic wrap. Set aside to cool to room temperature, or put in the fridge to cool more rapidly (this can keep in the fridge for a few days).
Next, add in the soft butter in thirds, waiting to incorporate the new butter until the previous amount is mostly mixed in, scraping the sides as you necessary. You want the butter to disappear into the dough, but not melt. Once the butter is added, knead for another 7-10 minutes, until the dough is smooth and springy and just a bit tacky.
Prepare a lightly floured work surface and scoop the dough gently out onto it. It pays to be gentle here, as we don't want to push out all the great air bubbles we have gotten in the first rise. I like to do 8 small rolls to a loaf pan (2 rows of 4), but you could easily do 6 (2 rows of 3), or even 4 (1 row of 4). In order to make sure my dough balls end up the same size, I like to weigh the dough at this point and calculate the size of balls (by weight). The last time, I ended up with roughly 16 (8 balls per loaf), 85 gram balls to form two loaves, but again, I recommend weighing and doing your math, as your overall dough mass may differ depending on if you did cup or weight measurements, etc.
Once you have all the balls separated and weighed, gently and ever so slightly flatten each one and fold the edges, and tuck underneath the dough to make a ball shape. You are looking to make the very center of the dough become the top, and the previous top pulled underneath. This should create a smoother top and a round ball. Once shaped, I like to place the rolls on the lightly floured work surface and gently cup over the top of them and move my hand in a circular motion gently rounding the balls and tightening the top of the dough more. Once you have all your balls shaped, gently place them into the two buttered 9-by-5-inch loaf pans and cover lightly with a cloth.
Let the loaves rest 30 to 45 minutes until the risen dough is ever so slightly peeking over the edge of the pan. Take your remaining egg and whisk with a teaspoon of water. Brush the tops of the bread with the egg wash and bake on the bottom 1/3 of oven until golden brown and puffed 35 to 40 minutes.
Let cool in the pan for about 10 minutes and then remove to a wire rack and let cool at least 1 hour, but more is better. Enjoy!