Season the chicken breasts with salt and pepper. Cook in a lightly oiled skillet over medium-high heat for 6–7 minutes per side, until the internal temperature reaches 165°F. Rest for 5 minutes, then slice against the grain.
Preheat the oven to 400°F. Arrange the wonton strips in a single layer on a baking sheet, brush lightly with oil, and bake for 5–6 minutes, flipping once, until deep golden and crisp. Let cool completely.
Whisk together the peanut butter, soy sauce, rice vinegar, honey, lime juice, sesame oil, garlic, ginger, and sriracha. Add warm water one tablespoon at a time, whisking after each addition, until the dressing coats the back of a spoon in a thin, even layer.
Shred the napa cabbage, julienne the carrots and bell pepper, and slice the green onions. Combine in a large mixing bowl.
Pour about three-quarters of the dressing over the vegetables and toss until evenly coated. Taste and add more dressing as needed.
Top with the sliced chicken, baked wonton strips, chopped peanuts, sesame seeds, and cilantro. Serve immediately.