In a small pot, whisk together the water and bread flour until no lumps remain.

Place the pot over medium-high heat, stirring continuously with a spatula or wooden spoon to prevent sticking and burning.

Continue to stir for 2 to 3 minutes until the mixture thickens. You will notice lines appearing on the bottom of the pot as you stir, indicating it's approaching the desired consistency.

Reduce the heat to low and continue stirring until the mixture becomes a thick, gel-like paste that holds its shape when lifted with a spatula without dripping or falling.

Transfer the prepared tangzhong to a small bowl and allow it to cool to room temperature for about 20 minutes.

In a microwave-safe bowl, combine the whole milk and water.

Microwave on high for approximately 15 seconds.

Add the granulated sugar to the warm milk mixture and stir until it is completely dissolved.

Microwave the mixture again for another 15 seconds.

Ensure the temperature of the liquid is lukewarm to the touch – if it feels hot, allow it to cool slightly before proceeding to avoid damaging the yeast.

Sprinkle the active dry yeast over the surface of the sugar mixture and let it stand for 5 to 10 minutes, or until it becomes foamy, indicating the yeast is active.

Weigh out 705 grams of bread flour into the bowl of your stand mixer, reserving the remaining 5 grams for dusting later if needed.

Add the cooled tangzhong, the prepared sugar/yeast mixture, and the room-temperature egg to the bowl with the flour.

Using the dough hook attachment, mix the ingredients gently on low speed (or speed setting 1) for about 2 minutes, or until the dough just comes together and starts to slightly adhere to the bottom of the bowl.

Cover the bowl with plastic wrap or a damp cloth and let the dough rest for 25 minutes. This resting period, known as the autolyse, allows the flour to fully hydrate and aids in gluten development.

After the resting period, resume kneading the dough on low speed (or speed setting 1).

Add one tablespoon of the softened or room-temperature unsalted butter to the dough every minute for the next 4 minutes. Ensure each addition of butter is mostly incorporated before adding the next.

Stop the mixer after 4 minutes, scrape down the sides of the bowl and the dough hook to ensure all the dough is being worked.

Continue kneading on low speed for another 4 minutes, adding the remaining butter one tablespoon at a time, again ensuring each addition is mostly incorporated before the next.

Repeat the process of stopping the mixer, scraping the bowl and hook, and then kneading for 4 minutes two more times, for a total of four 4-minute kneading intervals with butter additions.

Lightly grease a large bowl with oil or cooking spray.

Transfer the kneaded dough to the greased bowl, turning it once to coat all sides.

Cover the bowl tightly with plastic wrap or a damp cloth.

Let the dough proof in a warm place until it has doubled in size, which should take approximately 1 hour and 15 minutes. The exact time will depend on the ambient temperature of your kitchen.

Shaping and Second Proofing:

Once the dough has doubled, gently punch it down in the center to release the trapped gases.

Turn the dough out onto a very lightly floured work surface and divide it into two equal portions.

Place one portion of the dough in a lightly greased bowl, cover it, and transfer it to the refrigerator to proof more slowly. This portion can be baked later or the next day.

Take the remaining portion of dough and gently roll it out into a rectangle. Perform a letter fold by folding the top third of the dough down towards the center and then folding the bottom third up over the top.

Cover the folded dough with plastic wrap and let it rest for 20 minutes. This resting period allows the gluten to relax, making it easier to roll out.

After the rest, roll the dough out again into a long rectangle. Starting from one of the short ends, tightly roll the dough up into a cylinder. Tuck the ends underneath to create a neat loaf shape.

Place the shaped loaf, seam side down, into a greased bread loaf pan (approximately 9x5 inches).

Cover the loaf pan loosely with plastic wrap or a damp cloth and let the dough proof again in a warm place for about 45 minutes, or until it has nearly doubled in size and looks puffy.

At the 35-minute mark of the second proof, preheat your oven to 350°F (175°C). Position an oven rack in the bottom third of the oven.

Once the second proof is complete and the oven is preheated, place the loaf pan on the prepared oven rack.

Bake for 26 minutes, or until the crust is golden brown and the internal temperature of the bread reaches approximately 190-210°F. Using a food thermometer inserted into the center of the loaf is the most accurate way to determine doneness.

Immediately after removing the bread from the oven, rub the top crust generously with softened or melted unsalted butter. This will contribute to a softer crust.

Carefully remove the bread from the loaf pan and place it on a wire rack to cool completely for at least 3 hours before slicing and serving. This allows the internal structure to set, resulting in a better texture.

Note on the Second Dough Portion: The refrigerated portion of the dough can be baked within the next day. Allow it to come to room temperature for about 30-60 minutes before shaping and proceeding with the second proof and baking instructions.