Prepare the tangzhong by whisking the bread flour and milk together in a microwave-safe measuring cup. Microwave in 30-second increments, whisking well after each, until the mixture thickens into a smooth paste. In a 1,000-watt microwave, this takes about 1 minute 30 seconds.
3 tbsps (24 g) bread flour, ½ cup (121 g) whole milk
Transfer the warm tangzhong to the bowl of a stand mixer. Add the room-temperature milk and sugar and stir until well mixed. Sprinkle the active dry yeast over the mixture and let stand for about 5 minutes, until visibly foamy. If the yeast does not foam, discard the mixture and begin again with fresh yeast.
⅔ cup (147 g) whole milk, ¼ cup (49 g) granulated sugar, 2¼ tsps (7 g or 1 packet) active dry yeast
Fit the stand mixer with the dough hook. Add the bread flour and mix on low speed, level 2 on a KitchenAid, until a dough forms and begins releasing from the sides of the bowl.
3¼ cups (420 g) bread flour
Add the beaten egg and continue mixing until fully incorporated.
1 large egg
With the mixer running on low, add the very soft butter 1 tablespoon (14 g) at a time, allowing each addition to become fully incorporated before adding the next. Continue mixing until the dough begins pulling away from the sides of the bowl again.
4 tbsps (58 g) unsalted butter
Add the salt and knead with the dough hook for 10 to 15 minutes, until the dough is soft, smooth and slightly tacky. Check for sufficient gluten development using the windowpane test: a small piece of dough should stretch thin enough for light to pass through without tearing.
1½ tsps (8 g) fine sea salt
Lightly oil a large bowl. With damp hands, transfer the dough to the bowl and cover with plastic wrap or a clean kitchen towel. Let rise in a warm place for about 1 hour, or until doubled in size. Use the increase in volume rather than time as the final indicator.
neutral oil
Turn the dough onto a lightly floured surface and divide into 12 equal portions, approximately 70 grams each.
Shape each portion into a smooth ball by pinching the dough together underneath, then pulling the ball toward you against the lightly floured surface to create tension across the top.
Lightly grease a 9 x 13-inch (23 x 33 cm) baking dish (oil or cooking spray) and arrange the 12 shaped rolls evenly inside.
Cover and let rise for about 1 hour, or until the rolls have doubled in size.
Position an oven rack in the center of the oven and preheat to 350°F (177°C).
Whisk the egg and water together with a fork until evenly mixed. Brush the risen rolls lightly with the egg wash.
1 large egg, 1 tbsp (15 ml) water
Bake for 20 to 23 minutes, until the tops are golden brown and the center of the rolls registers 190°F (88°C) on an instant-read thermometer.
Let the milk bread rolls cool in the pan for about 10 minutes before serving.