Whisk bread flour, milk, and water in a small saucepan until smooth. Cook over medium-low heat while whisking constantly for 3-5 minutes until thickened to a smooth paste. Cover with plastic wrap pressed onto the surface and cool to room temperature.
Warm 1/3 cup milk to 100-110°F and pour into stand mixer bowl. Sprinkle yeast over the surface and let sit undisturbed for 5-10 minutes until foamy.
Add cooled tangzhong, bread flour, sugar, salt, and egg to the yeast mixture. Mix with dough hook on low speed for 2-3 minutes until a shaggy dough forms, then knead on medium for 5 minutes.
With mixer running, add softened butter in 2-3 additions. Knead for 8-10 minutes until dough is smooth and passes the windowpane test. The dough should be soft and slightly tacky.
Shape dough into a ball and place in a greased bowl. Cover and let rise in a warm spot for 60-90 minutes until doubled in size.
Punch down dough and divide into 3 equal portions. Roll each into an oval, fold long edges toward center, then roll up tightly from one short end. Place seam-side down in a greased 9x5 loaf pan.
Cover and let rise for 45-60 minutes until dough puffs about 1 inch above the rim of the pan.
Preheat oven to 350°F. Brush top of risen loaf with milk and bake for 30-35 minutes until internal temperature reaches 190°F and top is golden brown.
Remove from oven and rest in pan for 10 minutes. Turn out onto wire rack and cool completely for at least 1 hour before slicing.