In a small saucepan, whisk together the 3 tbsp flour, 1/2 cup water, and 1/4 cup milk.

Cook over medium heat, stirring constantly, until the mixture thickens into a smooth paste. This should take 3-5 minutes.

Once thickened, remove from heat and let it cool to room temperature.

In a large mixing bowl (or stand mixer with a dough hook), combine the bread flour, sugar, salt, and yeast.

Add the warm milk, egg, and the cooled Tangzhong.

Mix until a rough dough forms.

Add the softened butter and continue kneading for about 10-15 minutes, until the dough becomes smooth and elastic. It should pass the "windowpane test" (stretch the dough and it forms a thin, translucent sheet without breaking).

Shape the dough into a ball and place it in a greased bowl. Cover with plastic wrap or a damp towel and let it rise in a warm place for about 60-90 minutes, or until it doubles in size.

Once the dough has doubled in size, punch it down to release the gas.

Divide the dough into three equal portions. Roll each portion into a ball.

Roll out each ball into an oval shape, fold the short edges in towards the center, then roll up the dough into a cylinder (like a jelly roll).

Place the rolled pieces seam-side down into a greased loaf pan.

Cover and let the dough rise again for about 45 minutes, or until it reaches about 1 inch above the edge of the pan.

Preheat your oven to 350°F (180°C).

Brush the top of the loaf with an egg wash (1 beaten egg with a splash of water or milk) for a golden, shiny crust.

Bake for 25-30 minutes, or until the top is golden brown and the bread sounds hollow when tapped.

Remove from the oven and let the bread cool in the pan for a few minutes before transferring it to a wire rack to cool completely.