Activate the yeast: In a small bowl, combine warm milk, warm water, and a teaspoon of sugar. Sprinkle active dry yeast over the mixture and let it sit for about 5-10 minutes until frothy and bubbly, indicating that the yeast is active.
Mix the dough: In a large mixing bowl or the bowl of a stand mixer fitted with a dough hook attachment, combine 4 - 4.5 cups of bread flour, sugar, and salt. Make a well in the center of the dry ingredients and pour in the yeast mixture, softened butter, and beaten egg.
Knead the dough: Mix the ingredients together until a rough dough forms, then knead the dough on a lightly floured surface or in the stand mixer for about 10-15 minutes until smooth and elastic. The dough should be soft and slightly sticky but not overly wet.
First rise: Shape the dough into a ball and place it in a greased bowl. Cover the bowl with plastic wrap or a clean kitchen towel and let the dough rise in a warm, draft-free place for about 1-2 hours, or until doubled in size.
Shape the dough: Once the dough has doubled in size, gently punch it down to release any air bubbles, then divide it into equal portions. Shape each portion into a ball or loaf and place them in greased loaf pans or on a baking sheet lined with parchment paper.
Second rise: Cover the shaped dough with plastic wrap or a clean kitchen towel and let it rise again in a warm, draft-free place for about 30-60 minutes, or until doubled in size.
Preheat the oven: While the dough is rising, preheat your oven to 350°F (175°C).
Bake the bread: Once the dough has doubled in size, bake it in the preheated oven for about 25-30 minutes, or until golden brown and cooked through. If you're unsure, you can check for doneness by tapping the bottom of the bread – it should sound hollow.
Cool and enjoy: Remove the bread from the oven and let it cool in the pan for a few minutes before transferring it to a wire rack to cool completely. Once cooled, slice and enjoy your homemade Japanese milk bread!