To make carrots, place the water, vinegar and sugar in a small pot. Simmer for 30 seconds over medium-heat and stir to dissolve the sugar. Remove from the heat and cool to room temperature.
Place the carrots in a fine sieve. Add the salt and toss to coat the carrots with it. Set the sieve over a bowl or sink and let stand at room temperature for 30 minutes.
When the 30 minutes are up, rinse carrots with cold water and then pat dry. Place the carrots in a small bowl or jar. Pour in the vinegar mixture. Cover and refrigerate this pickle until needed below. The pickle can be made a day before needed.
To make the bao, in the bowl of a stand mixer fitted with a dough hook, combine the water, yeast, sugar and 2 tsp. oil. Let yeast mixture stand 5 minutes. Now add the flour and baking powder and knead on medium speed 6 minutes, until you have a smooth, fairly soft dough.
Set the dough in a lightly oiled, deep bowl. Cover with plastic wrap, and let rise at room temperature until doubled in size, about 75 minutes. Cut 16, 4” by 4” squares of parchment paper and divide and set them on 2 large baking sheets.
When risen, place the dough on a lightly floured surface and cut in half. Cut each half dough into 8 equal pieces. Shape each piece into a round ball.
Use a floured rolling pin to roll out each ball, one at a time, into an oval shape about 4” wide and an 1/8” thick. Lightly oil the top of the oval. Lightly coat the end of chopstick with oil. Place the oiled chopstick in the centre of the oval. Fold the dough over the chopstick, and then slowly pull out the chopstick. Set the folded dough on one 4” by 4” pieces of parchment paper.
Roll, oil and fold the remaining dough balls in this fashion. Cover the folded buns on the baking sheets with clean tea towels and let rise until doubled in size, about 60 to 75 minutes.
Set a steaming rack in a wok or large skillet or pot filled with water to level about 1” below the rack. Set over medium-high heat and bring water to a boil. Steam the buns, in batches and covered, for 8 minutes, until puffed up and cooked. Set buns back on the baking sheets and set aside until needed below.
To make the chicken filing, combine its first 7 ingredients in a bowl. Cover and marinate in the refrigerator 30 minutes.
Heat 3 Tbsp. of the oil in a wok or wide skillet set over medium-high heat. When very hot, add the chicken, including the marinade, and stir-fry until just cooked through, about 3 minutes. Add the bell pepper, prawns and chestnuts and stir-fry 2 minutes more.
Combine chicken sauce ingredients in a small bowl, and then add it to the pan. Cook until the sauce thickens and coats the chicken. Sprinkle with the 2 sliced green onions, and then take off the heat.
To serve, fill each bun with a little lettuce or cabbage, and then stuff with the chicken mixture and some cilantro. If desired, top the chicken with a little hoisin and hot sauce, and then dig in.