Whisk together 1/3 cup (43g) bread flour, 1/2 cup (120ml) milk, and 1/4 cup (60ml) water until smooth. Cook over medium-low heat while whisking constantly for 2-3 minutes until thickened to a pudding-like paste. Transfer to a bowl, cover with plastic wrap directly on the surface, and cool to room temperature.
In a stand mixer bowl with dough hook, combine 2 3/4 cups (350g) bread flour, sugar, salt, and instant yeast. Add the cooled tangzhong, egg, and 1/3 cup (80ml) milk. Mix on low speed for 1 minute until a shaggy dough forms, then increase to medium and knead for 8-10 minutes until smooth and pulling away from the bowl.
With the mixer on medium-low, add the softened butter cubes one at a time, waiting until each is absorbed before adding the next. Continue kneading for 5-7 minutes until the dough passes the windowpane test—stretching into a thin translucent membrane without tearing.
Shape the dough into a smooth ball and place in a lightly oiled bowl. Cover with plastic wrap and let rise in a warm spot (75-80°F / 24-27°C) for 60-75 minutes until doubled in size. Press a floured finger into the dough—it should spring back slowly and partially.
Gently punch down the dough and turn it out onto a floured surface. Divide into 3 equal pieces (about 200g each). Roll each piece into an oval, fold the long sides inward, and roll up tightly into a log. Place the logs seam-side down side by side in a greased 9x5 inch loaf pan.
Cover the pan loosely with greased plastic wrap and let rise again in a warm spot for 45-60 minutes until the dough crests just above the rim of the pan. Avoid over-proofing—the dough should spring back slightly when poked.
Preheat oven to 350°F (175°C). Whisk the egg yolk with 1 tablespoon milk and brush gently over the top of the loaf. Bake for 30-35 minutes until deep golden brown and internal temperature reaches at least 190°F (88°C). Tent with foil in the last 10 minutes if browning too quickly.
Let the loaf rest in the pan for 5 minutes, then turn out onto a wire rack to cool completely for 1-2 hours. Resist slicing warm to avoid a gummy texture—the crumb needs time to set properly.