Prepare the squash filling: In a bowl, combine the mashed butternut squash, chili crisp, chopped scallions, grated ginger, minced garlic, and soy sauce. Mix thoroughly and taste, adjusting seasoning if necessary.
Set up your dumpling station: Arrange the wonton wrappers, beaten egg, filling, and a small bowl of water on your work surface. Keep the wrappers covered with a damp cloth to prevent drying out.
Fill each dumpling: Place about 1 tablespoon of filling in the center of each wrapper. Brush the edges with beaten egg or water.
Fold and seal the dumplings: For a simple fold, bring all edges to the center and pinch to seal, creating a small purse shape. Alternatively, fold in half to create a half-moon and pleat the edges. Ensure there are no air pockets.
Start the broth: In a large pot, combine broth, soy sauce, ginger slices, smashed garlic, and star anise. Bring to a simmer and cook for 15 minutes to infuse flavors.
Strain and finish the broth: Remove the ginger, garlic, and star anise. Add rice vinegar and sesame oil, then keep warm over low heat.
Cook the dumplings: Bring a separate pot of water to a gentle boil. Carefully add dumplings in batches, being careful not to overcrowd. Cook until they float to the surface and the wrappers become translucent, about 3-4 minutes.
Serve immediately: Using a slotted spoon, transfer dumplings to serving bowls. Ladle the hot ginger soy broth over them. Garnish with scallions, cilantro, an extra drizzle of chili crisp, and a sprinkle of sesame seeds.