Bring a pot of water to boil on the stove. Use a ladle and gently lower the quail eggs into the boiling water. Boil for 2-3 minutes. Transfer eggs to a bowl of cold water. Let cool and peel. In a bowl, hydrate the dried Shitake mushrooms with two cups of boiling water. Put a plate on top. Set aside for 20 minutes. Then rinse in cold water, squeeze out any excess water, and chop finely.
In a large mixing bowl, stir together the milk, sugar, and vegetable oil. Add the bag of banh bao flour and stir until its well-blended. Cover with a kitchen towel and let sit for 15 minutes.
Slice the Chinese sausage diagonally. Cook it over medium heat in a nonstick pan until it releases its oil. Set aside.
In a mixing bowl, add the ground pork, chopped Shitake mushrooms, minced shallot and garlic, oyster sauce, salt and pepper. Mix together until well-blended. (This is easiest if you use your hands.)
After the 15 minutes of resting are done, move the dough to a large board and knead for five minutes. Roll the dough into a cylinder with your hands. Cut into 12 pieces. Cut the parchment paper into 12 squares about 3" x 3".
Roll each piece of dough into a ball and flatten using a rolling pin. Add a spoonful of the ground pork mix in the center, surrounded by a few pieces of Chinese sausage, and one quail egg.
Bring opposite edges of the dough together. Stretch the dough a little if you need to. Make sure all of the filling is covered inside. Pinch the edges together and twist so that the bun is closed.
Place each bao on a square of parchment paper. Arrange the baos on your bamboo steamer tray, leaving space in between each.
In a pan large enough that the bamboo steamer to sit on top (or inside), fill the bottom of the pan halfway with water. Be careful not to overfill and get the bottom tray of the steamer wet. Add 1/4 cup white vinegar (if you want the buns to be white, optional). Bring the pan water to a boil.
Add the steamer tray. Steam the bao on medium heat for 25 minutes.
Remove the bao from the steamer tray. Enjoy immediately, or you can easily store for later. They will be good in the fridge for 3-4 days, or in the freezer for two months.To reheat in the microwave, wrap in a damp paper towel and cook on high for 45 seconds. If they're still cold, add 15 more seconds.(In our house, they did not last for more than half a day. They were gobbled up very quickly.)