For the mushrooms:
1 tablespoon vegetable oil
1 heaping cup fresh shiitake mushrooms
1 tablespoon minced shallots
1 tablespoon minced fresh ginger
3 cloves garlic, minced
1/3 cup Shaoxing wine or dry sherry
1 cup low-sodium chicken broth
1/3 cup dark soy sauce
1/3 cup packed dark brown sugar
For the filling:
2 tablespoons cornstarch
1 tablespoon Shaoxing wine or dry sherry
1 ground ground turkey, preferably dark meat
1/2 cup thinly sliced scallions
2 tablespoons minced fresh ginger
2 tablespoons minced garlic
2 tablespoons minced lemongrass
1 tablespoons soy sauce
1 teaspoon toasted sesame oil
1/2 teaspoon five-spice powder
40 square wonton wrappers
Cornstarch, for dusting
1 large egg, beaten
For the broth:
8 cups (2 quarts) low-sodium chicken broth
2 tablespoons toasted sesame oil
2 tablespoons rice vinegar
1/4 cup thinly sliced scallions, for garnish
Braise the mushrooms: Heat the oil in a wok or large frying pan over medium-high heat. Add the mushrooms and sauté until lightly browned, 3 to 4 minutes. Add the shallots, ginger, and garlic and sauté until fragrant, 2 to 3 minutes. Add the wine or sherry and cook until reduced by half.
Stir in the remaining ingredients and bring to a simmer. Reduce the heat to low and continue simmering for 20 to 30 minutes.
Remove the mushrooms from the braising liquid to a medium bowl and refrigerate until cold, about 30 minutes. Strain the braising liquid through a fine-mesh strainer into a medium bowl and reserve for the broth.
When the mushrooms are chilled, cut into small dice (it should yield about a cup) and reserve for the filling.
Make the filling: Mix the cornstarch and Shaoxing in a large bowl until the cornstarch is dissolved. Add the chopped mushrooms and remaining filling ingredients and mix well to combine. Refrigerate until cold.
Wrap the wontons: Lightly dust a baking sheet with cornstarch. Place 1 teaspoon of the filling in the center of a wonton wrapper (keep the other wrappers covered with a damp towel to keep them from drying out). Brush the edges of the wonton with the beaten egg. Turn the wonton so it is facing you like a diamond. Take the bottom point of the wonton and fold it over until it touches the top point. Press the edges together, making sure to get rid of any air pockets. (It should now look like a triangle with the tip pointing up.)