Make tangzhong: whisk 3 tbsp flour with 1/2 cup milk in a small saucepan over medium-low heat for 2 to 3 minutes until thickened into a smooth paste. Spread on a plate and cool to room temperature.
Combine bread flour, sugar, yeast, and salt in a stand mixer bowl. Add warm milk beaten with egg, then add cooled tangzhong. Mix on low 2 minutes until shaggy.
Knead on medium 5 minutes. Add softened butter one tablespoon at a time, waiting for each to incorporate. Knead 8 to 10 more minutes until smooth and the dough passes the windowpane test.
Flatten dough, scatter chocolate chips across the surface, fold over and knead by hand 1 to 2 minutes to distribute. Shape into a ball, place in an oiled bowl, cover, and let rise 60 to 90 minutes until doubled.
Divide into 9 equal pieces. Shape each into a tight round ball and place in a greased 9x9 pan. Cover and let rise 40 to 50 minutes until puffy and pan-filling. Preheat oven to 350°F.
Brush tops with whole milk. Bake at 350°F for 20 to 24 minutes until deep golden and internal temp of center bun reaches 190°F. Tent with foil after 15 minutes if tops brown too fast.
Brush immediately with melted butter when out of the oven. Cool in pan 10 minutes before pulling apart and serving.
