Place the chicken breasts, chicken broth, soy sauce, sesame oil, rice vinegar, brown sugar, minced garlic, and grated ginger into the slow cooker.
Cover and cook on high for 3 to 4 hours or on low for 6 to 8 hours until chicken is tender.
Remove the chicken from the slow cooker and shred it using two forks, then return the shredded chicken to the slow cooker.
Add the julienned carrots, sliced mushrooms, and dried ramen noodles to the pot, ensuring the noodles are fully submerged in the hot liquid.
Cover and cook for an additional 5 to 10 minutes or until the noodles are tender and the vegetables are slightly softened.
Ladle the soup into bowls and garnish with sliced green onions, soft-boiled eggs, and sesame seeds before serving.