First, in a saucepan, combine flour and milk. Cook over low heat while stirring continuously until it thickens into a smooth, pudding-like paste. Remove from heat and let it cool completely before using.
Next, in a large bowl, combine flour, sugar, yeast, and salt. Then add eggs, milk, and the cooled tangzhong. Mix until a sticky dough forms.
Now begin kneading the dough until it becomes smoother. Gradually add the soft butter and continue kneading until the dough is soft, elastic, and passes the windowpane test.
Transfer the dough to a bowl, cover it, and let it rise for 1½–2 hours or until doubled in size.
Divide the dough into 8 equal portions (about 100g each). Shape each piece into tight, smooth balls and place them evenly spaced on a baking tray.
Cover the shaped buns and let them proof for 30–45 minutes until puffy. Avoid overproofing to maintain structure.
Gently brush the tops with egg wash made from yolk, milk, and sugar for a shiny, golden finish.
Preheat oven to 175°C (350°F) and bake for 15–18 minutes until deep golden brown.
Remove from the oven, optionally brush with butter, and let the brioche buns cool completely before slicing or serving.