Heat 1 tablespoon sesame oil in a large skillet over medium-high heat until shimmering.
Add the chopped shiitake mushrooms, shredded cabbage, shredded carrots, grated ginger, and minced garlic; cook, stirring occasionally, until the vegetables have softened and released their liquid, about 5 minutes.
Stir in 2 tablespoons soy sauce and the chopped chives and toasted sesame seeds; cook another 2–3 minutes until most of the liquid has evaporated. Remove from heat and let the filling cool slightly.
Spoon about 1 tablespoon of filling onto the center of each wonton wrapper. Brush water around the wrapper edge, fold into a half-moon, and pinch the edges tightly to seal. Repeat with remaining wrappers and filling.
Heat the remaining 1 tablespoon sesame oil in a large skillet over medium-high heat. When shimmering, arrange potstickers in a single layer, flat side down, and cook until the bottoms are light golden brown, about 2–3 minutes.
Carefully pour 1/4 cup water into the pan and immediately cover with a tight-fitting lid. Reduce heat to medium and steam the dumplings for 3 minutes, then remove the lid and cook 30–60 seconds more if needed to crisp the bottoms. Serve immediately with the sauce.
Meanwhile, whisk together 1/3 cup rice vinegar, 1/4 cup toasted sesame oil, 1–2 tablespoons gochujang or chili sauce, 2 tablespoons low-sodium soy sauce, and 1 tablespoon honey in a small bowl to make the toasted sesame honey soy sauce.