In a small saucepan over medium heat, whisk together bread flour, milk, and water until the mixture thickens and becomes smooth.
Remove from heat and let it cool to room temperature.
Brioche Dough
In a glass bowl, whisk together warm milk (about 100–105°F / 38–40°C), sugar, and yeast. Cover with plastic wrap and let it sit for 10 minutes until foamy.
In a stand mixer bowl, combine bread flour and salt.
Using the paddle attachment on low speed, slowly pour in the yeast mixture.
Add the cooled tangzhong and mix until combined.
Add the eggs and continue mixing on low speed until everything is fully incorporated.
Switch to the dough hook attachment and, on low speed, add butter in three additions.
Knead on low speed for about 10 minutes until the dough is smooth and elastic.
Transfer the dough to a greased glass bowl, cover with a kitchen towel and then plastic wrap, and place it on a heating pad for about 2 hours, or until doubled in size.
Punch down the dough to release the air.
Lightly flour your work surface and place the dough on it. Using a bench scraper, divide the dough into 8 equal pieces.
Shape each piece into a ball and place them on a baking sheet lined with a silicone mat.
Cover with another baking sheet and let the dough rise on the heating pad for 1 hour.
Preheat the oven to 375°F / 190°C.
Brush the tops of the buns with egg glaze.
Bake for 15–18 minutes, until golden brown.
Remove from the oven, brush with melted butter, and top with sesame or poppy seeds if desired.
Transfer to a wire rack and let cool completely for about 1 hour.
Glaze
In a small bowl, whisk together the egg and water.
Notes
It is essential to let brioche buns to cool completely, for about an hour. Otherwise, the texture of the buns will be doughy.