Make Tangzhong: Whisk water and 3 Tbsp flour in a small saucepan. Cook 2–4 minutes over medium heat, whisking constantly, until thick like mashed potatoes. Transfer to stand mixer bowl and let cool 15 minutes.
Activate yeast: In a small bowl, mix warm milk and active dry yeast. Let sit 5 minutes.
Combine dough: In the stand mixer bowl, add dry ingredients: bread flour, sugar, and salt. Add the cooled tangzhong, yeast–milk mixture, and 1 egg. Mix on low speed until a shaggy dough forms.
Knead dough: Knead on low for 10 minutes, adding butter 1–2 pieces at a time, until smooth and elastic.
First rise: Lightly oil dough and bowl, cover, and let rise 1 hour or until doubled in size.
Shape dough: Punch down and divide into 3 equal pieces. Shape each into a rectangle, fold, and roll into logs. Place seam-side down in pan.
Second rise: Cover and let rise 30–40 minutes or until doubled.
Bake: Preheat oven to 350°F. Bake 30–35 minutes until golden. Cool slightly before slicing.
