Cut the bak kwa into 16 pieces, about 1 x 1 inch square, or whatever shape you want really.
Prepare tangzhong roux:
Place milk and the bread flour in a small saucepan. Whisk until there is no more lumps. Cook this mixture over low-medium heat until it reaches 65 C (149 F). Remove from the heat and cover with a plastic wrap, touching the tangzhong so the skin won't form. I highly suggest using this the next day instead of using it on the same day. Store this in the fridge
Prepare the dough:
Mix all the dry ingredients together in a mixing bowl. Make a well in the center and pour in the wet ingredients and the tangzhong you made earlier. Use a dough hook attachment to knead the dough until all the crumbly dough starts to come together into one mass, it may take about 2-3 minutes for this to happen. Don't be tempted to add any liquid. You may need to stop halfway and scrape the dough off the hook and the sides of the bowl and knead again several times
Once it comes into a dough, stop the machine. The dough will not be smooth, don't worry. Cover the bowl with a plastic wrap and let it rest for 20 minutes
After the rest, turn on and knead the dough again and knead for 1-2 minutes until it is a bit smoother. Then add the salt and sugar into the dough. Continue to knead until the dough is relatively smooth. You need to stop and scrape the dough off the hook and the bowl several times during this process. I promise it's worth the effort!
Once you have a smooth dough, turn the machine back on again to knead and gradually add in the softened butter a bit by a bit. The dough will turn into a REAL sticky mess again because of the butter, just keep kneading and stop halfway and scrape the dough off the hook and bowl again and continue to knead until the butter is absorbed by the dough. IT WILL HAPPEN, don't worry! DO NOT be tempted to add any extra flour. You will get a shiny smooth dough that is very elastic, or what a baker calls a "windowpane stage". When you stretch the dough thin, it doesn't break
1st proofing:
Place this dough in a lightly oiled large bowl, cover it with a clean and damp tea cloth or plastic wrap and let it rise at a warm place for about 1 hour until the dough doubles in size
Shaping:
Punch the dough down, divide it into 16 smaller doughs. Work with one dough at a time and keep the rest covered
Slightly flatten the dough and then pull the edge to the middle to form a smooth dough ball. Place the seam side down. Place in a standard-size muffin liner. Take one piece of bak kwa and push it down, don't be afraid to do so. It will puff back up again later when you proof the buns. Repeat with the rest of the dough and bak kwa pieces
2nd proofing:
Put them on a baking tray and cover them with a clean cloth and let them rise in a warm place. I use my oven bread proof function and let them rise for about 1 hour or when you gently press the dough, it will slowly bounce back. If it bounces back immediately, proof a bit longer
Preheat oven to 350 F (180 C) 10 minutes before the buns are done proofing. Brush the surface of the buns with some egg wash. Drizzle with some kewpie mayo on top of each bun
Baking:
Place the baking tray in the middle rack and let them bake for 15-20 minutes or until they are golden brown on top
Remove from the oven and let them cool down in the pan for about 5 minutes and then lift the bread out and let it cool completely on a cooling rack. Enjoy and once it's cooled down completely, individually wrap it up with a plastic wrap to keep them soft and it's good for about 3 days or so