Whisk soy sauce, broth, rice vinegar, brown sugar, chili-garlic sauce, and cornstarch until smooth; set aside.
Cook noodles according to package directions, drain well, and keep nearby.
Heat half the oil in a large nonstick skillet over medium-high heat. Arrange potstickers flat-side down and sear until golden.
Pour in the water, cover, and steam until the potstickers are cooked through. Transfer to a plate.
Add remaining oil to the skillet. Stir in garlic and ginger; cook until fragrant, about 30 seconds.
Toss in the coleslaw mix and stir-fry until just wilted and crisp-tender.
Add the drained noodles. Pour the sauce over top and cook, tossing, until glossy and slightly thickened.
Return potstickers to the pan and fold gently to coat. Drizzle with sesame oil.
Finish with green onions and sesame seeds. Add a squeeze of lime juice and serve hot.