Cook noodles according to package instructions. Drain and set aside.
Heat 1 tbsp oil in a large skillet or wok over medium-high heat. Add the chicken, season lightly with salt, and cook until browned and cooked through. Remove and set aside.
In the same pan, add the remaining oil. Stir-fry the broccoli, bell pepper, carrots, and the white part of the green onions for 3–4 minutes until just tender.
In a small bowl, whisk together soy sauce, honey, brown sugar, rice vinegar, and sesame oil. In a separate bowl, mix cornstarch with water to make a slurry.
Return the chicken to the skillet. Add garlic and ginger and sauté for 1 minute. Pour in the sauce and bring to a simmer. Stir in the cornstarch slurry and let the sauce thicken.
Add the cooked noodles and toss everything together until coated evenly with sauce. Cook for another 2 minutes. Top with the green part of the onions before serving.