To make the dressing, place the peanut butter, soy sauce, rice vinegar, chili-garlic sauce, brown sugar, ginger, red pepper flakes, and 3 tablespoons of chicken broth together in a blender or bowl of a food processor. Blend until smooth. Season to taste with salt and pepper. Thin the dressing to your taste by adding more chicken broth or water.
Bring a large pot of lightly salted water to a boil. Add the linguine and cook until al dente, 8 to 10 minutes. Drain and place pasta into a large mixing bowl.
Add the chicken, carrots, green onions, red pepper, celery, and cilantro to the bowl with the linguine. Pour the dressing over the noodle-chicken mixture and toss until mixture is evenly coated. Divide the salad among eight serving plates, and sprinkle peanuts over each serving.