Make the tangzhong: Whisk flour + milk in a small saucepan. Cook over medium-low heat, stirring constantly. When it thickens to a smooth paste (≈65 °C / 150 °F), remove from heat. Scrape into a bowl, cover, and cool to lukewarm.
Mix the dough: In a bowl or stand mixer, combine main flour, salt, sugar, and yeast. Add tangzhong, remaining milk, and egg. Mix until shaggy.
Knead: Knead 10–12 minutes until gluten forms. Add butter gradually, kneading well between additions. Final dough should be very smooth, elastic, soft and slightly tacky (but not sticky).
First rise: Place dough in a lightly oiled bowl, cover, and let rise for 1–1½ hours, until doubled.
Shape: Gently deflate the dough, roll into a tight log, and place seam-side down in a well-buttered Pullman tin. Butter the lid too.
Second rise: Place the lid on (not closed) and let rise for 45–75 minutes, until dough is ~1 cm below rim. Slide lid fully closed just before baking.
Bake: Preheat oven to 180 °C (350 °F) and bake for 40–45 minutes. Internal temp target: 93–95 °C / 200–203 °F.
Cool: Unmould immediately and cool for at least 90 minutes before slicing.