Make the Tangzhong: In a small saucepan, whisk together the flour and milk until smooth. Cook over medium heat, whisking constantly, until the mixture thickens and becomes a smooth paste. This usually takes about 2-3 minutes. Remove from heat and let it cool completely.
Proof the Yeast: In a small bowl, combine warm water (105-115°F) with the yeast and 1 teaspoon of sugar. Let it sit for 5-10 minutes, or until foamy.

Combine Dough Ingredients: In the bowl of a stand mixer fitted with the dough hook, combine the tangzhong, proofed yeast mixture, remaining sugar, salt, egg, melted butter, and 2 cups of the flour. Mix on low speed until a shaggy dough forms.
Knead the Dough: Increase the speed to medium and knead for 5-7 minutes, or until the dough is smooth and elastic. If the dough is too sticky, add the remaining 1/4 cup of flour, 1 tablespoon at a time, until it comes together.
First Rise: Lightly grease a large bowl with oil. Place the dough in the bowl, turning to coat with oil. Cover with plastic wrap and let rise in a warm place until doubled in size, about 1-1.5 hours.
Shape the Buns: Punch down the dough and divide it into 9 equal pieces. Shape each piece into a smooth ball.
Second Rise: Place the buns in a lightly greased baking sheet. Cover with plastic wrap and let rise again for about 30-45 minutes, or until doubled in size.
Bake: Preheat oven to 375°F (190°C). Brush the tops of the buns with the milk. Bake for 15-20 minutes, or until golden brown.
Cool: Let the buns cool on a wire rack for a few minutes before serving.
Tips:

Use high-quality bread flour for the best results.
Make sure the milk for the tangzhong is warm, but not hot.
Don’t overmix the dough.
Let the dough rise properly for the best flavor and texture.
Enjoy your delicious and fluffy milk buns! They are perfect for breakfast, lunch, or a snack.

I hope this recipe helps you make some fantastic milk buns! Let me know if you have any other questions.