Combine milk and butter in a saucepan and heat to 100 degrees F. If the milk is hotter than 100 degrees F let it cool before proceeding, so you don't kill the yeast.
Pour the warm milk into the bowl of a stand mixer. Add yeast to the mixing bowl and stir briefly, just to dissolve the yeast.
Add the sugar, egg white, flours, and salt to the mixing bowl. With the paddle attachment, mix the ingredients on medium-low speed until completely combined, about 2 minutes.
At this point you may need to add the warm water if the dough is too dry and all of the flour has not absorbed.
Add the warm water, one tablespoon at a time until the flour absorbs. You should not need to add more than 4 tablespoons unless you packed the flour when measuring.
Remove the paddle attachment and put the dough hook on.
Knead the dough on low to medium speed for about five minutes. It should become smoother and pull away from the sides of the bowl.
If you do not have a dough hook you can drop the dough onto a lightly floured surface and knead it by hand for ten minutes.
After kneading by machine or by hand the dough should be tacky, but not overly sticky. You may need to add a little extra flour if the dough is too sticky to handle.
Turn out dough and form it into a ball. Place the ball in a large oiled bowl, flip over the oil the opposite side. Cover with plastic wrap. Place in warm spot and let rise until doubled in size, about one to two hours.
Remove the dough to a lightly floured surface.
Using a kitchen scale, divide the dough into 2 oz pieces and form them into balls.
Place the balls of dough back into the oiled bowl and cover with a damp towel. One at a time, remove the balls and roll them out flat with a rolling pin into an oval shape.
Brush the tops of the dough lightly with oil and then fold in half. Place each bun on a small square of parchment paper. Arrange the buns on a large tray, loosely covered with plastic wrap.
Allow the formed buns rise for another 15-20 minutes.
During the second rise, prepare a wooden or metal steamer by filling the bottom portion with water.
Arrange buns in the steamer so they are not touching, cover, and steam for 10 minutes. You may need to adjust the time depending on your particular steamer.
Open one bun to test it. If the inside is still raw, steam for a little longer.
If the top surfaces of the buns are wrinkled, then they have steamed for too long. Reduce the time for the next batch.
Serve buns immediately or store in the refrigerator and briefly re-steam to serve. Fresh is best!