Step 1: Make the Tangzhong

1. In a small saucepan, combine 3 tablespoons of bread flour with ½ cup milk and ¼ cup water.
2. Cook over medium heat, stirring constantly, until the mixture thickens into a smooth paste (about 2-3 minutes).
3. Remove from heat and let it cool to room temperature before using.

Step 2: Prepare the Dough

1. In a large mixing bowl, combine bread flour, sugar, salt, and instant yeast.
2. Add the cooled tangzhong, warm milk, and egg. Mix until a shaggy dough forms.
3. Knead the dough (by hand or with a stand mixer) until smooth, about 5 minutes.
4. Add the softened butter and continue kneading for another 8-10 minutes until the dough becomes soft and elastic.

Step 3: First Rise

1. Shape the dough into a ball and place it in a greased bowl. Cover with a damp towel or plastic wrap.
2. Let it rise in a warm place for 1-1.5 hours or until doubled in size.

Step 4: Shaping the Loaf

1. Once the dough has risen, punch it down to release air bubbles.
2. Divide it into three equal portions and roll each into a ball.
3. Roll out each ball into an oval, then fold it like a letter. Roll it up into a log shape.
4. Place the rolled pieces into a greased loaf pan.

Step 5: Second Rise

1. Cover the loaf pan and let the dough rise for another 40-50 minutes until puffy and nearly doubled in size.

Step 6: Bake the Bread

1. Preheat the oven to 350°F (175°C).
2. Brush the top with egg wash for a golden crust.
3. Bake for 25-30 minutes until the top is golden brown and the loaf sounds hollow when tapped.

Step 7: Cool & Serve

1. Remove the loaf from the pan and let it cool on a wire rack before slicing.
2. Enjoy warm with butter, or use for sandwiches and toast!