Make the tangzhong: In a small saucepan over low heat, combine the flour and milk and cook, whisking constantly, until thickened to a paste, 2 to 3 minutes. Immediately transfer the paste into a small bowl, scraping the sides of the saucepan with a flexible spatula; let cool until warm, 5 to 10 minutes. Texture should resemble mashed potatoes.
Make the milk bread: In a clean or new small saucepan, scald the milk over medium heat, bringing the milk to a gentle simmer (watch carefully as milk tends to boil over). Pour milk into a small bowl and cool until warm to the touch (about 110°F [42C]). Stir in yeast and a pinch of sugar and set aside until the surface of the mixture is foamy, 5 to 10 minutes.
In the bowl of an electric mixer fitted with a dough hook, combine the sugar, 335 grams bread flour, salt, and egg. Add the tangzhong and milk and mix on low until shaggy. Add the softened butter one piece at a time, mixing until fully incorporated before adding the next. Increase the speed to medium-high and continue to knead the dough until it is tacky and slightly sticky, 8 to 9 minutes. Transfer the dough to a lightly floured work surface. Wet your hands to prevent the dough from sticking, pinch and pull the ends of the dough to form a smooth ball.
Coat a large mixing bowl with 1 teaspoon of oil. Add the dough to the bowl, gently turning it to cover with oil. Cover the bowl with plastic wrap and set in a warm spot to proof until doubled in size, about 2 hours (or place in the refrigerator to proof for at least 8 hours or overnight).