Make the tangzhong. In a small saucepan, whisk the bread flour (30 g / 3 Tbsp) and milk (150 g / ⅔ cup) over medium-low heat, whisking constantly, until it thickens into a smooth paste, about 3–4 minutes. Remove from heat and let cool to room temperature.

Mix the dough. In a stand mixer bowl, combine the cooled tangzhong, bread flour (380 g / 3 cups), milk (120 g / ½ cup), egg, sugar (50 g / ¼ cup), and instant yeast (7 g / 2¼ tsp). Mix on low with the dough hook until a shaggy dough forms.

Knead in the butter. Add the salt (6 g / 1 tsp), then knead on medium 5 minutes. Add the softened butter (43 g / 3 Tbsp) one piece at a time, kneading until fully absorbed, about 8–10 more minutes, until the dough is smooth and passes the windowpane test.

First rise. Shape into a ball, place in a greased bowl, cover, and let rise in a warm spot 1–1½ hours, until doubled.

Divide and shape. Punch down the dough and divide into 12 equal pieces (about 65 g each). Shape each into a tight ball.

Second rise. Arrange the rolls in a buttered 9x13-inch pan, cover, and let rise 45–60 minutes, until puffy and touching.

Egg wash and bake. Preheat the oven to 350°F. Brush the rolls with beaten egg. Bake 18–20 minutes, until deep golden brown and the internal temperature reads about 190°F.

Cool. Brush with melted butter if desired, and cool at least 15 minutes before serving.