In a small saucepan, whisk together the bread flour, milk, and water until no lumps remain.

Place the saucepan over medium heat and cook, stirring constantly, until the mixture thickens to a pudding-like consistency. This should take about 5 minutes. Remove from heat and let cool to room temperature.

In a mixing bowl, combine the flour, sugar, salt, and active dry yeast.

Add the cooled Tangzhong, warm milk, and egg to the dry ingredients.

Using a stand mixer fitted with a dough hook, knead the dough on low speed until it comes together.

Add the softened butter and continue kneading for about 10-15 minutes until the dough is smooth and elastic.

Place the dough in a lightly greased bowl, cover, and let it rise in a warm place for about 1 hour or until doubled in size.

Punch down the risen dough and divide it into three equal parts.

Roll out each part into a rectangle and roll it up from the short side like a jelly roll.

Place the rolled pieces into a greased loaf pan.

Cover and let it rise for another 40 minutes or until doubled in size.

Preheat the oven to 350°F (175°C).

Brush the top of the risen dough with the egg wash mixture (1 egg beaten with 1 tbsp of milk).

Bake for 30-35 minutes or until the top is golden brown and the bread sounds hollow when tapped.

Let cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.