Place the chicken and all marinating ingredients into a medium bowl and mix thoroughly by hand until well coated.
In a small bowl, whisk together all sauce ingredients until fully combined.
Cook the noodles just until tender according to package directions, rinse with cold water to stop cooking, drain well, and set aside. For fresh noodles, dip briefly in boiling water for about 1 minute, then drain thoroughly.
Heat one tablespoon of oil in a large skillet on medium-high heat until shimmering. Arrange the chicken in a single layer with minimal overlap and cook undisturbed until the bottom is lightly browned, about 30 to 45 seconds. Flip and cook the opposite side until chicken turns opaque, another 30 to 45 seconds. Remove chicken and place on a plate.
Add the remaining tablespoon of oil along with ginger and garlic to the pan. Stir a few times to release their aromas.
Add carrots and cabbage to the pan and stir-fry for about a minute until the vegetables just begin to soften. Then add the noodles and toss a few times with tongs.
Return the cooked chicken to the skillet and pour the sauce evenly over all the ingredients. Toss everything together using tongs to coat well.
Include the sliced pepper and green onions, then toss once more until the sauce absorbs and the dish is heated through. Serve immediately while hot.