In a small saucepan, whisk together 1/4 cup whole milk, 3 tablespoons of bread flour, and 1/4 cup of water until smooth. Cook over medium heat for 2-4 minutes, whisking frequently until it thickens into a paste. Remove from heat and cool for 15 minutes.

Transfer the cooled tangzhong to a stand mixer bowl. Add 2 3/4 cups bread flour, 1/2 cup warm milk, 1/4 cup sugar, 1 teaspoon salt, 1 envelope of yeast, and 1 large egg. Stir until a shaggy dough forms.

Attach the dough hook and knead on low speed for about 5 minutes. Gradually add cubed butter and continue kneading for another 5 minutes until the dough is smooth and elastic.

Shape the dough into a ball, oil the top, and cover with plastic wrap. Let it rise in a warm place for 40-50 minutes or until doubled in size.

Grease a loaf pan. Punch down the dough to remove excess air, divide into three equal pieces, shape into rectangles, and roll into logs. Place logs in the pan with seams down.

Cover the loaf and let it rise again until doubled, about 30-40 minutes. Preheat oven to 350°F.

Mix the remaining egg with 1 tablespoon of milk or water. Brush this mixture over the risen loaf.

Place the loaf in the preheated oven and bake for 32-35 minutes until golden brown. Cool in the pan for a few minutes before transferring to a wire rack to cool completely.