Whisk water, milk, and flour in a saucepan until smooth.
Cook over medium-low heat, stirring constantly, until thick and glossy.
Cool completely.
Mix milk, water, sugar, and yeast.
Rest 5–10 minutes until foamy.
Add egg and tangzhong. Mix well.
Add flour and salt. Stir until dough forms.
Knead until smooth, then add butter gradually.
Continue kneading until dough is silky, elastic, and slightly tacky.
Cover loosely and let rise at a warm place until doubled, 45–60 minutes.
Divide into 3–4 pieces and place into greased 9×5-inch loaf pan.
Cover loosely and let rise until dough just rises above pan rim, 25–35 minutes.
Whisk egg yolk, milk, and sugar. Gently brush over the risen dough.
Bake uncovered at 350°F (175°C) for 30–35 minutes until golden.
Cool 10 minutes in pan, then remove.
