Make Tangzhong: In a small saucepan, whisk together milk and flour; 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 look like mashed potatoes.) Transfer to a small bowl and let cool until an instant-read thermometer registers 130°F (54°C) or until room temperature. Makes about ⅓ cup. (See note 2 for tip)
Make dough: In the bowl of a stand mixer, whisk together 1⅔ cups (208 grams) flour, sugar, yeast, and salt. Add warm milk, eggs, and Tangzhong; using the paddle attachment, beat at low speed until combined, about 1 minute. With mixer on low speed, gradually add remaining 1 cup (125 grams) flour, beating until a shaggy dough forms; scrape sides of bowl.
Switch to the dough hook attachment. Beat at medium-low speed until dough becomes smooth, elastic, and slightly tacky, about 8 to 11 minutes. (Dough should pull away from sides of bowl.) With mixer on medium-low speed, add butter, 1 tablespoon (14 grams) at a time, beating until combined after each addition (7 to 8 minutes total). Increase mixer speed to medium, and beat until a smooth, elastic dough forms, about 10 minutes. Reduce mixer speed to low; add chocolate and cherries, beating until evenly distributed.
First rise: 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, about 1 to 1½ hours.
Line baking sheets with parchment paper.
Shape buns: Turn out dough onto a clean surface, and punch down dough. Press dough to a 10x8-inch rectangle. Divide dough into 20 portions (about 45 grams each). Using the palm of your hands, gently roll each portion into a smooth ball. Place 3 to 4 inches apart on prepared pans.
Final rise: Cover with lightly greased plastic wrap, and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, about 1 hour.
Preheat oven to 375°F (190°C).
Egg wash: In a small bowl, whisk together water and egg; gently brush over tops of risen buns.
Bake: Bake until golden brown and an instant-read thermometer inserted near center registers 190°F (88°C), 8 to 10 minutes. Let cool on pans for 5 minutes. Remove from pans and let cool completely on a wire rack.