In a small saucepan, whisk together the water, milk and flour. Cook over medium heat for several minutes, whisking constantly, until it thickens into a paste.

Remove from the heat, scrape the tangzhong into a small bowl, and refrigerate to cool it down while you make the dough.

In the same saucepan you used to make the tangzhong (no need to wash the pan first), melt the butter over medium heat. Stir in the milk, and warm the mixture just until it reaches between 110-115 degrees Fahrenheit.

Remove from the heat. Stir in the yeast and 1 tablespoon of sugar. Let sit for a few minutes until it starts to get bubbly.

In the bowl of your stand mixer, combine all but ½ cup of the flour with the remainder of the sugar, the warm yeast mixture, the cooled tangzhong, salt and egg.

With the dough hook, knead the dough for 10 minutes, gradually adding the remaining ½ cup of flour, as needed. The dough should be soft and smooth, and will wrap around the dough hook, but may still stick to the bottom of the bowl. Don't be tempted to add more flour, though, or the bread will be dense.

Scrape the dough into a greased bowl. Cover tightly with plastic wrap, and set in a warm place to rise until doubled in size. This could take anywhere from 30-90 minutes, depending on the freshness/brand of yeast, and how warm your kitchen is.If your oven has a bread proof setting, you can use that. Otherwise, preheat your oven to the lowest setting, turn it off, then set the dough inside to rise.

Turn the dough out onto a lightly floured surface, and lightly flour the top of the dough, too. Use a bench scraper to cut the dough into 4ths, then shape each section into a ball.

Use a rolling pin to roll each ball of dough into an oval measuring 4x8 inches, then roll the oval up into a log. Repeat with all 4 balls of dough, until you have 4 logs of dough.

Butter and flour the inside of a Pullman loaf pan, and arrange the 4 logs of dough side by side in the pan, with the seams of the logs facing down against the bottom of the pan.

Cover the pan tightly with plastic wrap, and set in a warm place to let the dough rise again, just until it has nearly reached the top of the pan. This may take 20-30 minutes. Remove the plastic wrap.

Preheat the oven to 350 F, and position a rack in the center of the oven.

For the egg wash, whisk together the egg and milk, then use a pastry brush to brush the egg wash over the bread. This will give the bread a beautiful golden brown shine as it bakes.

Bake the bread for about 35-40 minutes, until golden brown, and an instant digital read thermometer inserted into the center of the bread reads between 190-195 degrees Fahrenheit.

Let the bread cool in the pan for 5 minutes, then turn it out onto a cooling rack. Cool for 30-45 minutes, then enjoy warm, or at room temperature.