Tangzhong Whisk together water, milk, and flour in a small saucepan. Cook over medium heat, whisking constantly, until it thickens into a smooth, pudding-like paste. Transfer to a bowl and press plastic wrap directly onto the surface. Let cool to room temperature.
Make the Dough Warm the milk until lukewarm (about 37°C — slightly warm, not hot). Stir in about 5 g of the sugar and the yeast. Let sit for 5–10 minutes until foamy.
In a large bowl, combine: Cooled tangzhong, Egg, Remaining sugar, Flour, Salt, Yeast mixture Mix until a slightly sticky dough forms. Use a stand mixer with a dough hook or knead by hand. Knead in the Butter Knead the dough for about 5 minutes. Add the softened butter. Knead for another 5 minutes until the dough becomes smooth, elastic, and slightly glossy. It should feel soft but not sticky.
First Rise Shape the dough into a ball. Cover and let rise for 1–2 hours, until doubled in size.
Shape + Second Rise Divide the dough into 9 equal pieces. Roll each piece into a smooth ball. Place in a greased or parchment-lined 20 × 20 cm baking pan. Cover and let rise again for 45–60 minutes, until nicely puffed.
Bake Preheat oven to 175°C. Beat the egg with the milk and brush the rolls gently. Bake in the center of the oven for about 20 minutes, until golden brown. If they brown too quickly, loosely cover with foil during the last few minutes.
Finish Let cool in the pan for a few minutes. Brush with melted butter. Sprinkle with flaky salt if desired and enjoy.
