Prepare Ingredients: Slice the chicken breast, julienne the carrots, thinly slice the green onions (separate white and green parts), mince the ginger, thinly slice the cabbage, and slice the mushrooms.
Cook Chicken: In a large pot, heat sesame oil over medium-high heat. Add the chicken slices and cook until no longer pink, about 5 minutes.
Add Aromatics and Veggies: Add the minced ginger and the white parts of the green onions. Sauté for 1-2 minutes until fragrant. Add the carrots, cabbage, and mushrooms. Cook for another 3-4 minutes until the vegetables start to soften.
Create Broth: Pour in the chicken broth, soy sauce, and mirin. Bring to a boil.
Cook Noodles: Add the noodles to the pot and cook according to package instructions (usually about 4-5 minutes).
Finish Soup: Taste and adjust seasoning with salt and pepper if needed. Stir in the green parts of the green onions. Ladle the soup into bowls and enjoy hot.