"This is a delicious recipe I got from my mamasan when I lived in Okinawa. Everyone raves over them. Store extra dipping sauce in the refrigerator up to 1 week and use to dip chicken or other meats or vegetables."
Heat 1 teaspoon sesame oil in a skillet over medium heat; add curry paste. Cook until aromatic, about 30 seconds. Add cabbage, carrot, and 1 1/2 tablespoons garlic; cook and stir until cabbage is wilted and moisture has evaporated, about 3 minutes. Remove from heat; allow to cool.
Mix pork, egg, 1/4 cup green onions, water chestnuts, ginger, soy sauce, fish sauce, 2 teaspoons sesame oil, and brown sugar together in a bowl. Stir in cabbage mixture.
Whisk together rice vinegar, low-sodium soy sauce, 1/3 cup green onions, 1 teaspoon sesame oil, 1 clove garlic, and chili paste. Let dipping sauce flavors meld for at least 15 minutes before serving.
Separate and place gyoza wrappers onto your work surface. Spoon about 2 teaspoons of the filling onto the center of each wrapper. Use your finger or a pastry brush to lightly moisten edges of wrappers with water. Fold one side of the wrapper over the filling onto the opposite side to form a crescent-shaped gyoza. Press edges together to seal.
Heat 1 tablespoon vegetable oil in a large skillet over medium heat. Cook half the gyoza until lightly browned, about 1 minute per side. Pour 1/4 cup broth into skillet; reduce heat and cover. Steam until broth evaporates and gyoza are translucent, about 5 minutes. Repeat with remaining oil, gyoza, and broth.