Place the sliced onion, mushrooms, minced garlic, and grated ginger in the bottom of the slow cooker. Spread them into a relatively even layer so the chicken sits on top rather than sinking into the liquid.

Lay the whole chicken breasts on top of the vegetables. Pour the chicken broth over everything, then drizzle in the soy sauce, sesame oil, and rice vinegar. Season with the salt and black pepper.

Place the lid on the slow cooker and cook on low for 5 to 6 hours. The chicken is done when it registers 165 degrees F on an instant-read thermometer and pulls apart very easily when pressed with a spoon.

About 45 minutes before the cook time is up, add the frozen edamame directly to the slow cooker and stir it into the broth. Replace the lid and let it cook through for that remaining time. Adding it late keeps the edamame firm and bright rather than soft and grainy.

Remove the chicken breasts and shred them using two forks, pulling the meat apart into pieces that are small enough to fit comfortably on a spoon. Return the shredded chicken to the slow cooker and stir to combine with the broth and vegetables.

Taste the broth and adjust salt or soy sauce if needed. The broth should taste savory, gingery, and slightly nutty from the sesame oil.

Ladle the stew into bowls over rice or rice noodles if you like, or serve it on its own. Top each bowl with sliced green onions and a light sprinkle of sesame seeds before serving.