In a small saucepan, combine water, milk, and 2 tablespoons flour for the tangzhong. Whisk to combine and cook over medium heat until the flour hydrates and thickens, whisking constantly. This takes about 1 minute. Continue cooking for an additional 1 ½ to 2 minutes to cook out the raw flour taste and thicken further. The mixture should be quite thick, like mashed potatoes.

Scrape the tangzhong into the bowl of a stand mixer.

To the mixer bowl with the tangzhong, add sugar, flour, salt, yeast, milk, butter, and one egg for the dough. I purposefully add them in this order to separate the hot tangzhong from the yeast and egg. By the time all the ingredients incorporate, it will have cooled and there's no need to wait before mixing.

Fit the stand mixer with the dough hook attachment and mix on the lowest speed until all the ingredients have incorporated and there are no more visible pieces of butter.

Increase mixer to medium speed and knead for 4 minutes, or until the dough pulls a smooth windowpane. Note: This dough is sticky and will stick to the bottom of the bowl. It should clean the sides of the bowl and not tear when the hook is raised from the bowl. This is normal!

Using a bowl scraper, scrape the dough into a large lightly oiled bowl. Cover and bulk proof for 1 ½ hours, or until doubled in size and light and airy.

Scrape the dough onto a clean work surface and divide it into 9 pieces for large rolls (approximately 124g each) or 12 pieces for standard dinner rolls (approximately 93g each). This recipe can also be divided in half and shaped into loaves just like sandwich bread.

Shape each piece into a roll by folding the edges into the center, flipping seam-side down, and using a push-pull motion to create a smooth ball. Place shaped rolls in a greased 9x9-inch square pan (for 9 rolls) or 9x13-inch rectangular pan (for 12 rolls).

Cover and final proof for 1 to 1 ½ hours, or until the tops of the rolls crest over the sides of the baking dish. Meanwhile, preheat the oven to 350°F conventional (no fan).

Gently brush the tops with beaten egg (egg wash).

Bake at 350°F for 25 minutes for 9 large rolls or 20-22 minutes for 12 smaller rolls, until the rolls are a nice golden brown on top. Don't overbake or the crust will become too thick!

Remove from oven and let cool slightly before serving. These are best served warm or at room temperature.