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, only if 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 buns 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 45-90 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 flour the top of the dough, too. Gently roll the dough out into a square, about ½ inch thick.

Cut the dough into 9 equal portions for large buns, or 12 portions for medium sized buns.

Take a piece of dough, cup it in your hand, and pinch the dough together to form it into a bun, so that it's smooth on top and pinched on the bottom.

Place the buns, smooth side facing up, into a greased 9-inch baking dish (or a 9x13 baking dish for 12 buns).Baker's Note: If you want to make the dough and shape the buns the night before a holiday, complete all the steps up to this point, cover the baking dish with plastic wrap, and refrigerate overnight. In the morning, proceed to the next step. The buns may need an hour or longer to come to room temperature and warm up before they start to puff and fill the pan. Once they do, they're ready to bake.

Cover the pan loosely with a clean kitchen towel, and set aside until the buns are starting to puff up, about 20-30 minutes.

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, and brush the egg wash over the buns. This will give them a beautiful golden brown shine as they bake.

Bake the milk buns for about 22-25 minutes, until golden brown on top and baked through.

Let cool for a few minutes, then pull the buns apart and serve warm.