Place the chicken breasts in the slow cooker. Add the chicken broth, ginger, garlic, onion, and carrots.
Cover and cook on low for 6 hours or until the chicken is tender enough to shred easily.
Remove the chicken, shred it with two forks, and return it to the slow cooker.
Stir the miso paste into 1 cup of the hot broth until fully dissolved, then pour the mixture back into the slow cooker.
Add the edamame and cook on high for 30 more minutes.
Ladle into bowls and top with sliced green onions.