The evening prior, combine the flour, water, and sourdough starter in a small bowl, stirring until no dry bits remain. Transfer this to a clean jar, loosely cover, and leave it to ferment in a warm place overnight until doubled in volume.
For the tangzhong, whisk together the flour and milk in a medium saucepan set over medium heat. Continue stirring constantly until the mixture thickens into a smooth paste. Remove from the heat and scrape the paste into the bowl of a stand mixer.
Pour the cold milk over the tangzhong to bring down its temperature. Add the granulated sugar, main dough flour, milk powder, egg, salt, and the fully ripe sourdough starter. Begin mixing on low speed until a rough dough forms.
Increase to medium speed and knead the dough for about 5 minutes until it starts to strengthen. Gradually incorporate the softened butter, adding a few pieces at a time, and continue kneading for an additional 10–12 minutes until the dough is elastic and pulls away cleanly from the sides of the bowl.
Allow the dough to rest for 5 minutes, then perform a windowpane test by gently stretching a small portion; it should stretch thin without tearing, indicating proper gluten development.
Place the dough in a lightly oiled bowl, cover, and allow it to rise in a warm area (around 77°F/25°C) for about 4 hours, or until it has expanded by roughly 50%. In cooler environments, create a warm, moist space by placing the dough in a switched-off oven with a bowl of hot water, refreshing the water as needed.
Once bulk fermentation is complete, refrigerate the dough for at least 30 minutes to firm it slightly, easing the shaping process.
Turn the dough out onto a lightly floured surface and gently deflate it. Weigh the dough and divide it into three equal portions, shaping each piece into a tight ball.
Using a rolling pin, flatten each ball into an elongated oval roughly 20cm (8 inches) long. Roll each oval up tightly, like a Swiss roll, then position the rolls seam side down into a greased or parchment-lined 9x5 inch loaf pan.
Allow the shaped dough to rise once more in a warm spot (around 77°F/25°C) until doubled in size, which typically takes 3–4 hours depending on ambient temperature.
Preheat the oven to 350°F (180°C). Beat an egg with a tablespoon of water and gently brush this egg wash over the surface of the dough.
Bake for approximately 30–35 minutes, or until the loaf is deeply golden brown on top. If the bread darkens too quickly, tent it loosely with foil. When fully baked, the loaf should sound hollow when tapped underneath.
Remove the bread from the pan and cool it completely on a wire rack before slicing.