To make Tangzhong (white roux), whisk bread flour, milk, and water together in a medium saucepan until smooth. Cook, stirring constantly, over medium heat until mixture thickens into the consistency of mashed potatoes, about 2 minutes. Place plastic wrap over the top of the roux, and allow to cool to room temperature for at least 15 minutes.
Place all the dry ingredients into a stand mixer bowl. Whisk together, then set the bowl on the mixer with a dough hook.
Whisk the milk and eggs together, then slowly add it to the cooled roux while whisking until smooth.
With the mixer on the second to lowest setting, slowly add the egg/roux/milk mixture to the dry ingredients.
With the mixer still running, start adding the softened butter one tablespoon at a time (8 tablespoons in total) until all the butter has been incorporated. Increase the speed to medium-low and knead until the dough is very strong and elastic, 10 to 12 minutes. Remember that this is a high hydration dough, so it will seem sticky.
Brush a large bowl with butter, then transfer the dough to the buttered bowl. Cover, and let rise for at least 1.5 hours or until it has doubled. In the meantime, coast 2 loaf pans (8½-by-4½-inch ) with melted butter (we like to use glass loaf pans).
Lightly flour a large board, or your counter. Gently punch down the dough, then turn it out onto the prepared counter. Using a chef ’s knife or bench scraper, divide the dough into 4 equal portions, about 12 ounces each. Form each into a ball. Take one ball, and gently pat it down, then stretch it into a square(about 6x6 inches). Fold by thirds, like a letter; pinch the ends, turn the dough seam side down, and place into one end of one of the prepared loaf pans. Do this with the remaining pieces - see the picture above. Cover the pans, and let rest for about 1 hour, or until the dough rises to the top of the pans.