Trim any excess fat from the chicken breasts. Cut the chicken into super thin sliced pieces. On a separate cutting board, cut the carrots into thin strips. Cut out the core of the cabbage and then cut the cabbage into thin strips and pieces as well. Cut the onion into thin slices, as well as the green onions. Fill a large soup pot with water and turn to high heat. Once the water has come to a boil, add the noodles and stir them around. Boil the noodles until al dente. Once the noodles are cooked, drain them and place them back in the same soup pot.
In a large saute pan, or in an electric skillet, heat half of the butter until melted. Add the chicken breast pieces to the butter and saute until the chicken is tender and cooked through. Season the chicken with salt and pepper. Place the chicken into the pot with the cooked noodles. In the same pan, heat the rest of the butter and add the onions and carrots to the pan. Saute until tender and then add the sliced green cabbage. Continue to saute the vegetables until they are tender and cooked. Season them with more salt and pepper. Add these vegetables into the pot with the noodles and chicken. In the same saute pan, add the chow mein sauce ingredients and whisk together while cooking the sauce on medium-low heat. Simmer the sauce until it thickens. Once the sauce is thickened, pour it into the chicken and noodles pot and toss to coat. Garnish the dish with the fresh, green onions that have been sliced. Serve warm.