Whisk 3 tbsp bread flour and 1/2 cup water in a small saucepan; cook over medium heat, stirring constantly, until thickened into a pudding-like paste, about 3 minutes. Let cool.

In a stand mixer with a dough hook, combine the cooled tangzhong, 1/2 cup milk, 1/4 cup sugar, yeast, and egg. Mix on low to break up the yeast. Add 2 1/2 cups bread flour and 1 tsp salt; knead on low for 3 minutes. Rest 5 minutes, then add 3 tbsp softened butter a piece at a time and knead on medium for 8 to 10 minutes until smooth and elastic.

Shape into a ball, place in a greased bowl, cover, and let rise until doubled, about 1 to 1.5 hours.

Punch dough down, divide into three pieces, roll each into an oval, fold sides in, then roll up into a cylinder. Place seam-side down in a greased 9x5-inch loaf pan.

Cover and let rise until dough crowns 1 inch above the pan, 45 to 60 minutes. Meanwhile, preheat oven to 350°F (175°C). Bake 30 to 35 minutes until golden and internal temp registers 190°F (88°C). Tent with foil if browning too fast.

Cool in the pan for 5 minutes, then turn out onto a wire rack to cool completely before slicing.