Combine broth, soy sauce, sugar, chili sauce, lime juice, and ginger and boil 5 minutes. Meanwhile, toss chicken with cornstarch until completely coated. Add chicken and shiitakes to broth and simmer 15 minutes. Remove the pan from the heat and stir in snap peas, red pepper, zest, and cilantro. Let soup stand 2 to 3 minutes before serving.