Special equipment: a deep-frying thermometer
Preheat the oven to 350 degrees F.
Slice the squash in half and place the halves face-up on a baking sheet. Bake until soft and fork-tender, 35 to 40 minutes. Scrape into a large bowl, add 2 tablespoons of the butter while still warm and mash with a fork until mostly smooth. (Alternatively, you can puree in a food processor.) Allow to cool.
Meanwhile, in a small saute pan over medium heat, add the oil and swirl to coat the bottom of the pan. Add the garlic and ginger and cook 1 minute to soften. Add the shiitake mushrooms and season with salt and pepper. Cook until soft, 3 to 4 minutes. Transfer the mixture to a plate to cool completely.
In the bowl containing the squash, fold in the cooled mushroom mixture and remaining 2 tablespoons butter and stir until well blended. Adjust the seasoning as necessary.
Place 1 wonton wrapper on a work surface with 1 corner nearest you. Place 1 scant tablespoon of the filling in the center of the wrapper, moisten the edges with the egg wash and fold the bottom half over the top to create a triangular dumpling. Bring the left and right sides under the dumpling, moisten the points with the egg wash and pinch together to seal. Repeat with the remaining wrappers and filling.
Fill a fryer or medium heavy pot one-third full with oil. Over high heat, bring the oil to 350 degrees F on a deep-frying thermometer. Add half the wontons and fry until golden brown, about 2 minutes. Remove with a large mesh spoon and drain on paper towels. Repeat with the remaining wontons. Season with salt. Transfer to a platter.
In a small skillet, warm the pomegranate seeds over high heat. Once the pan is hot and the seeds begin to sizzle, add the rum and light with a lighter to flambe. Let the alcohol flames burn off, then pour the seeds over the wontons.
Drizzle with the Pomegranate-Vinegar Syrup, sprinkle with the chopped chives and serve.
In a nonreactive saucepan on low heat, add the vinegar and pomegranate seeds. Bring to a simmer and reduce by three-quarters, about 20 minutes. Test the reduction by drizzling a little on a chilled plate; the syrup should hold a line when the plate is tilted.
Transfer the syrup to a small glass jar and cover when cooled. Store in the refrigerator up to 4 weeks.
Use extreme caution when igniting alcohol. Remove the pan from the heat source before adding the alcohol. Pour the alcohol into the pan and carefully ignite with a match or click lighter. Return the pan to the heat and gently swirl to reduce the flames.