Bring a large pot of salted water to a boil and cook the udon noodles according to package directions; drain and set aside.
In a bowl, whisk together tamari or soy sauce, chili paste, rice vinegar, lemon juice, honey, toasted sesame oil, chopped garlic, chopped ginger, and 1/4 cup water until smooth.
Heat 1 tablespoon oil in a large skillet over medium heat. Add the chicken, season with black pepper, and cook until no longer pink, about 4–6 minutes for thin slices or ground chicken.
Add the frozen stir-fry vegetables to the skillet and cook 2–3 minutes until heated through.
Slowly pour the sauce mixture into the skillet with the chicken and vegetables, add the chopped scallions, and bring to a low simmer.
Add the cooked udon noodles to the skillet and toss gently, cooking until the sauce evenly coats the noodles and everything is heated through, about 1–2 minutes.
Serve immediately topped with toasted sesame seeds and chopped cilantro or Thai basil.
