Prepare the water roux or Tangzhong first by whisking together the flour and milk in a saucepan and cooking it while stirring or whisking continuously over medium-low heat until it gets a pudding-like consistency. Let it cool down to room temperature before using.
Using a stand mixer with dough hook attachment, mix the flour, sugar and salt together. Add the yeast, Tangzhong, butter, egg, and lukewarm milk then mix for 12 minutes at medium-low speed. It will be very sticky at the beginning but just keep on beating until the dough is smooth and no longer too tacky.
Form the dough into a ball and coat it with oil. Place in a bowl and cover with a kitchen towel or plastic wrap. Let it rise until it almost doubles in size (30 mins-2 hours).
Gently punch down the dough and transfer on a surface greased with oil. Roll the dough in a rectangle, with a length almost thrice the length of the loaf pan. Then fold like a letter, from right covering the center then from left covering both layers. Then roll the dough jellyroll fashion from the end closest to you towards the farther end.
Place the dough in a 5"x9" loaf pan greased with oil with the seam facing down and cover with kitchen towel. Allow the dough to rise for the second time until it doubles in size.
Bake in a preheated oven at 180°C for 20 minutes with the pan placed on a lower or middle rack.
In a small bowl, beat 1 small egg and 1 tablespoon water until frothy. Take out the loaf from the oven and brush the exposed part with egg wash then place back in oven for to go on baking for another 10-15 minutes or until top is nice brown and shiny.
Take the loaf out of the oven and let it cool down for some minutes. Remove it from pan to cool down completely. Slice with a bread knife in your preferred thickness.