To prepare the filling, place a wok or stir-fry pan over high heat. Add 1 tablespoon of oil, swirling the wok or pan to coat the sides. Add the garlic and fry until fragrant for about 10 seconds. Add the cabbage, carrot, chives, and red onion and stir-fry until the vegetables are tender and crisp for about 2 minutes. Add the oyster sauce and sesame oil and toss to coat. Transfer the vegetable mixture to a colander to drain. Set aside in a medium bowl.
Place a medium non-stick skillet over medium-high heat until hot. Add the remaining 1 teaspoon of oil, swirling to coat the bottom. Pour the egg mixture into the pan, tilting the pan so the egg coats the bottom, and cook until the bottom of the omelette is set, or for about 1 minute. Season with salt and pepper. Turn the omelette over and cook for about 30 seconds until the other side is just set. Slide the omelette onto a cutting board. Fold in half and cut into thin shreds. Transfer to the bowl of vegetables and set aside until cool.
To form the dumplings, place 1 dumpling wrapper on a clean, flat surface, keeping the remaining wrappers covered with a damp kitchen towel to prevent them from drying out. Place 2 heaping tablespoons of filling in the center of the wrapper. Moisten the edges of the wrapper with the egg wash. Top with a second wrapper and secure the edges. Starting from one end of the wrapper, slightly fold over the edge (like a pie crust) and continue folding around the dumpling. Repeat this process with the remaining wrappers and filling. Keep the dumplings covered with a damp kitchen towel to prevent them from drying out.
To make the Yin dumplings, heat 2 tablespoons of the oil in a large, non-stick skillet over medium-high heat. Add half of the filled dumplings, flat side down and cook without turning until golden brown for about 3 minutes. Add the vinegar and water. Reduce the heat to medium, cover, and steam until the dumplings are tender, or for about 4 minutes.
For the Yang dumplings, heat the remaining 2 cups oil in a wok or stir-fry pan over medium-high heat until hot. Working with a few at a time, add the remaining uncooked dumplings and shallow-fry, turning once, until golden brown and crisp, or for about 2 minutes per side. Lift the dumplings out with a wire strainer or slotted spoon and drain on paper towels.
To make the Soy Vinegar Dipping Sauce, mix all the ingredients in a small bowl and stir until combined. Store in a covered container in the refrigerator. It will keep for up to one week. You can choose to add more jalapeño to make it spicier, or omit them entirely.
Transfer the Yin and Yang dumplings to a serving plate. Serve with the dipping sauce on the side.