Whisk bread flour, water, and milk for starter together in a small saucepan until very smooth. Place over medium-low heat and cook, stirring often, until very thick and just about to bubble. If you want to check, it should be at least 150 degrees F (65 degrees C). Pull off the heat and transfer into a measuring cup; you will need exactly 1/2 cup. Let cool to room temperature, about 30 minutes.
Prepare dough: Combine bread flour, cooled starter, salt, sugar, yeast, warm milk, and egg in the bowl of a stand mixer fitted with the dough hook attachment. Beat on low speed until dough begins to form a ball and pull away from the sides of the bowl, scraping down the sides as needed, about 5 minutes.
Stop the mixer and scrape dough off the hook and into the bowl. Add butter and knead on low speed until dough is slightly tacky but very smooth and elastic, scraping down the sides as needed, 10 to 12 more minutes.
Remove dough from the bowl and shape into a smooth ball. Transfer into a lightly buttered bowl, cover, and let rise in a warm spot until doubled in size, about 1 to 1 1/2 hours.
Preheat the oven to 350 degrees F (175 degrees C). Very generously butter a 9×5-inch loaf pan.
Transfer dough to a work surface (you don’t need flour.) Press and knead out all the air and shape into a 9-inch wide rectangle. Starting at one 9-inch side, roll dough lengthwise into a log. Transfer log into the prepared loaf pan with the seam down. Press on dough a bit to be sure it’s evenly distributed. Cover the pan and let dough rise until it comes to just about the top of the pan, 35 to 45 minutes.
Brush the top of the loaf with just enough milk to lightly coat the surface.
Bake in the center of the preheated oven until the top is golden brown and the load is beautifully puffed up, about 35 minutes. Remove from the oven and let rest in the pan for 10 minutes. Very carefully remove bread and place on a wire rack to cool all the way to room temperature before slicing, about 1 hour.
Try this bread on my Japanese Egg Salad Sandwich (Tamago Sando).
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