Make the dressing: Combine peanut butter, tamari or soy sauce, sesame oil (or olive oil), lime juice, grated garlic, chopped pickled ginger, chopped jalapeño (or Fresno), hot sauce, and honey in a food processor or blender; pulse until smooth. Thin with water, 1 teaspoon at a time, until desired consistency is reached.
Assemble the salad: In a large bowl, combine shredded cabbage, cooked shredded chicken, shredded carrot, chopped cilantro and/or basil, chopped mango, and chopped peanuts.
Toss with dressing: Pour about half the dressing over the salad and toss to coat; add more dressing to taste.
Finish and serve: Sprinkle with toasted sesame seeds and chopped green onions, top with crispy wontons, and serve with a squeeze of lime.