For tangzhong: In a small saucepan, whisk together milk and flour until well combined. Cook over medium-low heat, whisking constantly, until thickened, whisk leaves lines on bottom of pan, and an instant-read thermometer registers 150°F (66°C). (Mixture will resemble mashed potatoes.) Transfer to a small bowl, and let cool until an instant-read thermometer registers 130°F (54°C) or until room temperature.

For dough: In a small saucepan, heat milk over medium heat until scalding (an instant-read thermometer registers 180°F/82°C). Remove from heat, and let cool until an instant-read thermometer registers 110°F (43°C) to 115°F (46°C).

In a small bowl, whisk together warm milk, 1 teaspoon (4 grams) sugar, and yeast. Let stand until foamy, 5 to 10 minutes.

In the bowl of a stand mixer, whisk together 1½ cups (191 grams) flour, salt, and remaining 3 tablespoons plus 2 teaspoons (44 grams) sugar by hand. Add cooled tangzhong, yeast mixture, and 1 egg (50 grams). Using the paddle attachment, beat at low speed until well combined. With mixer on low speed, gradually add 1 cup (127 grams) flour, beating just until a shaggy dough forms, stopping to scrape bottom and sides of bowl.

Switch to the dough hook attachment. Beat at low speed until dough is tacky, elastic, and slightly smooth, about 8 minutes; add up to remaining ¼ cup (32 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. Addbutter, 1 tablespoon (14 grams) at a time, beating until combined after each addition (about 5 minutes total).

Scrape sides of bowl and dough hook. Beat at low speed until dough is completely smooth and passes the windowpane test (see Note), about12 minutes. Turn out dough onto a clean surface, and shape into a smooth round.

Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 45 minutes to 1 hour.

Line baking sheets with parchment paper.

Punch down dough; turn out onto a lightly floured surface. Divide dough into 12 portions (about 60 grams). Shape each portion into a smooth ball; cover and let stand for 5 to 10 minutes.

Place balls 3 to 4 inches apart on prepared pans. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 1 to 1½ hours.

Preheat oven to 350°F (180°C).

In a small bowl, whisk together 1 tablespoon (15 grams) water and remaining 1 egg (50 grams); brushonto dough.

Bake until golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), about 17 minutes. Let cool on pans for 10 minutes. Remove from pans, and let cool completely on a wire rack.