Heat 2 tablespoons of the oil in a large nonstick skillet over medium-high heat. Add the leeks and garlic and cook until leeks begin to soften, about 3 to 4 minutes. Add the cabbage, vinegar, soy sauce, and ginger and cook, stirring, until cabbage is soft, about 3 to 4 minutes more. Remove from the heat and stir in the carrot. Allow the filling to cool.
Transfer cabbage mixture to a mixing bowl and work together until fully incorporated. Keep refrigerated until ready to use.
To make the potstickers, lay a wonton wrapper on the work surface and put about 1 tablespoon of the filling in the center. Dip your finger in a little water and wet the edges of the wonton wrapper; this will help the potsticker seal when cooking. Bring the outside edges of the wonton wrapper together over the filling and press it together to form a pouch. Continue with the remaining wrappers and filling.
To cook the potstickers, heat the remaining oil in a very large skillet over high heat. Place all of the potstickers upright in the pan and cook, without disturbing them, until browned on the bottom, about 1 to 2 minutes. Add 1/2 cup water to the pan and immediately cover. Allow the potstickers to steam for about 3 minutes, then remove the skillet lid. Continue cooking until the water is completely evaporated.
Transfer to a platter and serve with the dipping sauce.
Asian Dipping Sauce:
Stir all ingredients together in a small bowl until sugar is dissolved. Refrigerate until ready to serve.