Step 1: Prepare the Tangzhong. In a small saucepan, whisk together 2 tablespoons bread flour and 1/2 cup water until smooth. Cook over medium-low heat, stirring constantly, until the mixture thickens to a paste (about 2-3 minutes). It should register 149°F (65°C) on an instant-read thermometer. Remove from heat, transfer to a small bowl, cover directly with plastic wrap to prevent a skin from forming, and let it cool completely to room temperature.
Step 2: In the bowl of a stand mixer fitted with the dough hook attachment, combine the 2 1/2 cups bread flour, granulated sugar, salt, and instant dry yeast. Mix briefly to combine.
Step 3: Add the warm milk, lightly beaten egg, and the cooled tangzhong paste to the dry ingredients. Mix on low speed until a shaggy dough forms, about 1-2 minutes.
Step 4: Increase the mixer speed to medium-low and knead for about 10 minutes, until the dough comes together and starts to pull away from the sides of the bowl. It will be sticky. Gradually add the softened butter, one piece at a time, allowing each piece to incorporate before adding the next. Continue kneading for another 10-15 minutes on medium speed, until the dough is smooth, elastic, and passes the 'windowpane test.'
Step 5: Lightly grease a large bowl. Transfer the dough to the bowl, turning once to coat. Cover the bowl with plastic wrap or a damp towel and let it rise in a warm place for 1 to 1.5 hours, or until doubled in size.
Step 6: Gently punch down the dough to release the air. Transfer to a lightly floured surface. Divide the dough into 3 equal pieces. Roll each piece into a rectangle (about 8x5 inches). Fold the top third down to the center, then the bottom third up, pressing to seal. Turn the dough seam-side down and roll it into a log, about the length of your loaf pan. Repeat with the remaining dough pieces.
Step 7: Arrange the three shaped dough logs side-by-side in a greased 9x5 inch loaf pan. Cover the pan loosely with plastic wrap or a damp towel and let it rise in a warm place for another 45-60 minutes, or until nearly doubled in size and the dough reaches the top of the pan.
Step 8: Preheat your oven to 350°F (175°C). If desired, brush the top of the risen dough with the egg wash (egg yolk mixed with milk) for a golden crust. Bake for 30-35 minutes, or until the top is golden brown and an instant-read thermometer inserted into the center reads 190-200°F (88-93°C).
Step 9: Remove the loaf from the oven and immediately transfer it to a wire rack to cool completely. Do not slice until fully cooled to ensure the best texture. Enjoy!